THE  QUANTUM  THEORY 


THE 
QUANTUM     THEORY 


• 

BY 


FRITZ   REICHE 

PROFESSOR  OF   PHYSICS  IN    THE   UNIVERSITY  OF  BRESLAU 


TRANSLATED    BY    H.    S.    HATFIELD,    B.Sc.,    Ph.D..    AND 
HENRY    L.    BROSE.    M.A. 


WITH    FIFTEEN    DIAGRAMS 


NEW    YORK 

E.    P.    BUTTON    AND    COMPANY 
PUBLISHERS 


66038 


PRINTED     IN     GREAT     BRITAIN     BY 
THE  ABERDEEN   UNIVERSITY   PRESS 


c 


o 

CONTENTS 

^  CHAP.  PAGE 

^             INTRODUCTION 1 

j,      I.  THE  ORIGIN  OF  THE  QUANTUM  HYPOTHESIS   ....  2 

k\       II.  THE  FAILURE  OF  CLASSICAL  STATISTICS          ....  13 

III.  THE  DEVELOPMENT  AND  THE  RAMIFICATIONS  OF  THE  QUANTUM 

THEORY.        * 16 

IV.  THE  EXTENSION   OF  THE   DOCTRINE  OF  QUANTA  TO  THE  Mo- 

^                      LECULAR  THEORY  OF  SOLID  BODIES         ....  29 

V.  THE  INTRUSION  OF  QUANTA  INTO  THE  THEORY  OF  GASES       .  68 

VI.  THE  QUANTUM  THEORY  OF  THE  OPTICAL  SERIES.     THE  DE- 
VELOPMENT   OF    THE    QUANTUM    THEORY    FOR    SEVERAL 

DEGREES  OF  FREEDOM 84 

VII.  THE  QUANTUM  THEORY  OF  RONTGEN  SPECTRA        .        .         .  109 

VIII.  PHENOMENA  OF  MOLECULAR  MODELS  i      .....  117 

)    IX.  THE  FUTURE         .        .        .      ' 125 

W           MATHEMATICAL  NOTES  AND  REFERENCES        ....  127 

A          INDEX    .                                                                                       .  181 


THE  QUANTUM  THEORY 


INTRODUCTION 

r~r"vHE  old  saying  that  small  causes  give  rise  to  great  effects 
J[  has  been  confirmed  more  than  once  in  the  history  of 
physics.  For,  very  frequently,  inconspicuous  differences  be- 
tween theory  and  experiment  (which  did  not,  however,  escape 
the  vigilant  eye  of  the  investigator)  have  become  starting- 
points  of  new  and  important  researches. 

Out  of  the  well-known  Michelson-Morley  experiment, 
which,  in  spite  of  the  application  of  the  most  powerful 
methods  of  exact  optical  measurement,  failed  to  show  an 
influence  of  the  earth's  movement  on  the  propagation  of 
light  as  was  predicted  by  classical  theory,  there  arose  the 
great  structure  of  Einstein's  Theory  of  Kelativity.  In  the 
same  way  the  trifling  difference  between  the  measured  and 
calculated  values  of  black-body  radiation  gave  rise  to  the 
Quantum  Theory  which,  formulated  by  Max  Planck,  was 
destined  to  revolutionise  in  the  course  of  time  almost  all 
departments  of  physics. 

The  quantum  theory  is  yet  comparatively  young.  It  is 
therefore  not  surprising  that  we  are  confronted  with  an 
unfinished  theory  still  in  process  of  development  which, 
changing  constantly  in  many  directions,  must  often  destroy 
what  it  has  built  up  a  short  time  before.  But  under  such 
circumstances  as  these,  in  which  the  theory  is  continually 
deriving  new  nourishment  from  a  fresh  stream  of  ideas  and 
suggestions,  there  is  a  peculiar  fascination  in  attempting  to 
review  the  life-history  of  the  quantum  theory  to  the  present 
time  and  in  disclosing  the  kernel  which  will  certainly  out- 
last changes  of  form. 
1 


CHAPTEE  I 
The  Origin  of  the  Quantum  Hypothesis 

§  i.   Black-Body  Radiation  and  its  Realisation  in  Practice 

THE  Quantum  Theory  first  saw  light  in  1900.  When,  in 
the  years  immediately  preceding  (1897-1899),  Lummer 
and  Pringsheim  made  their  fundamental  measurements  1  of 
black-body  radiation  at  the  ReichsanstaU,  they  could  have 
had  no  premonition  that  their  careful  experiments  would 
become  the  starting-point  of  a  revolution  such  as  has  seldom 
occurred  in  physics. 

In  the  field  of  heat  radiation  chief  interest  at  that  time  was 
centred  in  the  radiation  of  a  black  body  (briefly  called  "  black- 
body  "  radiation),  that  is,  of  a  body  which  absorbs  completely 
all  radiation  which  falls  on  it  and  which  thus  reflects,  trans- 
mits, and  scatters2  none.  We  may  shortly  call  to  mind 
the  following  facts.  It  is  known  that  any  body  at  a  given 
temperature  sends  out  energy  in  the  form  of  radiation  into 
the  surrounding  space.  This  radiation  is  not  energy  in  a 
single  simple  form  but  is  made  up  of  a  number  of  single 
radiations  of  different  colours,  i.e.  of  different  wave-lengths  A 
or  of  different  frequencies  3  v.  In  other  words,  it  forms  in 
general  a  spectrum  in  which  radiations  of  all  frequencies 
between  v  =  0  and  v  =  oo  are  represented.  Further,  these 
radiations  are  present  in  varying  "  intensities."  We  define 
this  term  thus.  Consider  the  radiation  emitted  from  unit 
surface  of  the  body  per_second  in  a  certain  direction  ;  break 
it  up  spectrally  and  cut  out  of  the  spectrum  a  small  frequency 
interval  dv  such  that  it  contains  all  frequencies  between  v  and 
v  +  dv.  The  energy  of  radiation  Ev  thus  sliced  out  (namely, 
the  emissivity  of  the  body  for  the  frequency  v)  may  be  defined 
in  the  following  terms  :  * 

Ev  =  ^Kvdv          ,        .         .     (1) 
2 


BLACK-BODY  RADIATION  3 

provided  that — as  we  shall  assume  for  the  sake  of  simplicity 
— the  surface. of  the  body  emits  uniform  and  unpolarised 
radiation  in"  all  directions. 

The  magnitude  K,,  thus  defined  is  called  the  intensity  of 
radiation  of  the  body  for  the  frequency  v.  It  is  in  general  a 
more  or  less  complicated  function  of  the  frequency  v,  of  the 
absolute  temperature  of  the  body  T,  and  of  the  inherent 
properties  of  the  body.  The  black  body  alone  is  unique  in 
this  respect.  •  For  its  radiation  and  therefore  its  Kv  is,  as 
G.  Kirchhoff*  was  the  first  to  point  out,  dependent  only 
on  the  frequency  .v  and  the  absolute  temperature  T,  that  is, 
mathematically, 

Kv=f(v,T)     ....     (2) 

This  formula  which  gives  the  relation  between  the  intensity 
of  radiation  from  a  black  body,  the  temperature,  and  the 
"  colour  "  is  called  the  radiation  formula  or  the  law  of  radia- 
tion of  a  black  body. 

To  calculate  this-  relationship  on  the  one  hand  and  to 
measure  it  on  the  other  were  unsolved  problems  at  that 
time.  Unimpeachable  measurements  were  of  course  possible 
only  if  one  could  succeed  in  constructing  a  black  body  which 
approached  sufficiently  near  the  theoretical  ideal.  This  im- 
portant step,  the  realisation  of  the  black  body,  was  taken  by 
0.  Lummer  and  W.  Wien*  on  the  basis  of  KirchhofTs1 
Law  of  Cavity  Eadiation,  which  states  :  In  an  enclosure 
or  a  cavity  which  is  enclosed  on  all  sides  by  reflecting  walls, 
externally  protected  from,  exchanging  heat  with  its  surroundings, 
and  evacuated,  the  condition  of  "  black  radiation  "  is  auto- 
matically set  up  if  all  the  emitting  and  absorbing  bodies  at  the 
walls  or  in  the  enclosure  are  at  the  same  temperature.  In  a 
space,  therefore,  which  is  hermetically  surrounded  by  bodies 
at  the  same  temperature  T  and  which  is  prevented  from  ex- 
changing heat  with  its  surroundings,  every  beam  of  radiation 
is  identical  in  quality  and  intensity  with  that  which  would  be 
emitted  by  a  black  body  at  the  temperature  T. 

Lummer  and  Wien,  therefore,  had  only  to  construct  a 
uniformly  heated  enclosure  wi.th  blackened  walls  having  a 
small  opening.  The  radiation  emitted  from  this  opening  was 
then  "  black "  to  an  approximation  which  was  the  closer 


4  THE  QUANTUM  THEORY 

the  smaller  the  opening,  that  is,  the  less  the  completeness  of 
the  enclosure  was  disturbed.  The  manner  in  which  the 
intensity  Kv  of  the  black  radiation  thus  realised  depended 
on  the  frequency  v  and  the  temperature  T  had  next  to  be 
determined.  The  above-mentioned  investigation  of  Lummer 
and  Pringsheim  was  devoted  to  this  purpose. 

§  2.  The  Stefan-Boltzmann  Law  of  Radiation  and  Wien's 
Displacement  Law 

While  experimental  research  was  proceeding  on  its  way, 
theory  was  not  idle,  for  valuable  pioneer  work  was  being 
done  inasmuch  as  two  fundamental  laws  were  set  up.  In 
the  first  place,  L.  Boltzmann*  proved,  with  the  help  of 
thermodynamics,  the  law  previously  enunciated  by  Stefan,9 
that  the  sum-total  of  the  radiation  from  a  black  body, 
taking  all  the  frequencies  together,  namely,  the  quantity 

K  =  \    K.vdv,  is  proportional  to  the  fourth  power  of  its  absolute 

Jo 
temperature  :  10 

K  =  y  .  T4  (y  =  const.)      .          .          .      (3) 

The  laws  proposed  by  Wien  n  entered  more  deeply  into  the 
question.  Wien  imagines  the  black  radiation  enclosed  in  a 
closed  space  with  a  perfectly  reflecting  piston  as  one  wall, 
and  then  supposes  the  radiation  to  be  compressed  adiabatically, 
as  in  the  case  of  gases  (that  is,  no  passage  of  heat  to  or  from 
the  cavity  is  allowed  during  the  process),  by  infinitely  slow 
movements  of  the  piston.  Now,  if  we  express  the  change 
which  this  process  causes  in  the  energy  of  a  definite  colour 
interval  dv  in  two  ways,  and  if  we  take  into  consideration 
that  the  waves  reflected  at  the  piston  undergo  a  change  of 
colour  according  to  Doppler's  principle,  we  succeed  in  limiting 
very  considerably  the  unknown  functional  dependence  of  the 
quantity  Kv  on  v  and  T.  There  is  thus  obtained  a  re- 
lation of  the  form  12 


in  which  c  is  the  velocity  of  light  in  vaciio,  the  function  F 
being  left  undetermined.  From  this,  Wien's  Displacement 
Law,  the  conclusion  13  may  be  drawn  that  the  frequency 


WIEN'S  LAW  OF  RADIATION  5 

I'max  for  which  1C  (plotted  as  a  function  of  v)  is  a  maximum 
is  displaced  towards  higher  values  proportional  to  T  as  the 
temperature  increases  : 

Vmax  =  COnst.  .  T  .  .  .       (4ft) 

If,  as  is  usual  in  physical  measurement,  we  use  the  wave- 
length A  =  -  instead  of  the  frequency  as  the  variable,  Wieris 

Law  assumes  a  somewhat  different  form.  For  if  we  consider 
the  radiant  energy  of  a  narrow  range  of  wave-length  d\  cor- 
responding to  the  frequency  range  dv,  and  write  it  in  the 

form  E^dX,  then  EidK  =  K^dv,  that  is  :  E^  -  K*>  .  ^.    In 

A 

place  of  (4)  and  (4a)  we  then  get  the  relations  : 


>-max  .  T  .  const.  =  S  .         .     (5a) 

§  3.  Wien's  Law  of  Radiation 

To  formulate  the  law  of  radiation  it  was  therefore  neces- 
sary only  to  evaluate  the  unknown  function  F  in  (4)  or  (5). 
But  this  was  just  the  central  point  of  the  whole  question, 
and  the  most  difficult  part  of  the  problem. 

Here,  too,  Wien  made  the  first  successful  attack.  On  the 
basis  of  not  entirely  unobjectionable  calculations,  which  were 
founded  on  Maxwell's  law  of  distribution  of  velocities  among 
gas  molecules,  he  arrived  at  the  following  specialised  form  14 
of  the  function  F  :  — 

p  —  a    e-pf  (a  and  ft  are  two  constants). 

Thus  the  law  of  radiation  (4)  assumes  the  form 

K,»a£.«-4    .         .  .  .      (6) 

which  is  called  Wien's  Law  of  Kadiation. 

How  far  did  experiment  confirm  these  theoretical  results  ? 
While  the  Stefan-Boltzmann  Law  and  Wien's  Displacement 
Law  were  confirmed  to  a  large  extent  by  the  observations  of 
Lunimer  and  Pringsheim,™  both  experimenters  found  Wien's 


6  THE  QUANTUM  THEORY 

Law  of  Eadiation  confirmed  only  for  high  frequencies,  that  is, 
for  short  wave-lengths  (more  precisely,  for  large  values  of 

m),  and  detected,  on  the  other  hand,  systematic  dis 


for  small  frequencies,  that  is,  for  long  wave-lengths.™  They 
maintained  with  unswerving  persistence  that  these  discrepan- 
cies were  real  in  spite  of  objections  from  authoritative  quarters. 
For  while  F.  Paschen 17  imagined  that  he  had  proved  by  his 
work  that  Wien's  Law  of  Radiation  was  universally  valid, 
Max  Planck,  in  his  detailed  theory  of  irreversible  processes 
of  radiation,18  had  arrived  again  at  Wien's  radiation  formula 
by  a  more  rigorous  method.  Starting  from  Kirchhoff's  Law 
of  Cavity  Radiation,  according  to  which  the  presence  of  any 
emitting  or  radiating  substance  whatsoever  in  a  uniformly 
heated  enclosure  produces  and  ensures  the  maintenance  of 
the  condition  of  black-body  radiation,  Planck  chose  as  the 
simplest  schematic  model  of  such  a  substance  a  system 
of  linear  electromagnetic  oscillators,  and  investigated  the 
equilibrium  of  the  radiation  set  up  between  them  and  the 
radiation  of  the  enclosure.  This  is  to  be  understood  as  fol- 
lows :  Each  of  the  Planck  oscillators — as  such  we  may,  for 
example,  assume  bound  electrons  capable  of  vibration — pos- 
sesses a  fixed  natural  frequency  v  and  responds,  on  account 
of  its  weak  damping,  only  to  those  waves  of  the  radiation  in 
the  enclosure  whose  frequencies  lie  in  the  immediate  neigh- 
bourhood of  v,  while  all  other  waves  pass  over  it  without 
effect.  The  oscillator  thus  acts  selectively,  as  a  resonator,  in 
just  the  same  way  as  a  tuning-fork  of  definite  pitch  com- 
mences to  sound  only  when  its  own  "  proper"  tone,  or  one 
very  near  it,  is  contained  in  the  volume  of  sound  which  strikes 
it.  In  this  process  of  resonance,  however,  the  oscillator  ex- 
changes energy  with  the  radiation  inasmuch  as,  on  the  one 
hand,  it  acts  as  a  resonator  in  abstracting  energy  from  the 
external  radiation,  and,  on  the  other,  it  acts  as  an  oscillator 
and  radiates  energy  by  its  own  vibration.  Hence  a  dynamic 
equilibrium  is  set  up  between  the  oscillator  and  the  radiation 
of  the  enclosure,  and,  indeed,  between  just  those  waves  of 
the  radiation  which  have  the  frequency  v.  In  this  state  of 
equilibrium  the  radiation  of  frequency  v  acquires  an  intensity 
Kt.  which,  according  to  Kirchhoff's  Law,  is  equal  to  the  intensity 


THE  QUANTUM  HYPOTHESIS  7 

of  black-body  radiation  at  this  temperature.  Secondly,  the 
energy  U  of  the  oscillator  passes  in  the  course  of  time  through 
all  possible  values,  the  mean  value  19  U  of  which  is  found  to 
be  proportional  to  the  intensity  Kv,  a  result  which  seems  im- 
mediately plausible  since  the  excitation  of  the  oscillator  will 
be  greater  the  more  intense  the  radiation  that  falls  on  it. 
The  exact  calculation  of  this  relationship  between  Sv  and  U 
on  the  basis  of  classical  electrodynamics — this  is  the  first 
part  of  Planck's  calculations — leads  to  the  fundamental 
formula : 

*,=£•*?        -         -         •         '     (7) 

In  the  second  part  Planck  *  determined  U,  although  by  a 
method  that  is  not  free  from  ambiguity,  as  a  function  of  v  and 
T  on  the  basis  of  the  second  law  of  thermodynamics.  He 
obtained 

ff-«*-4      ....   (8) 

The  combination  of  (7)  and  (8)  gives  us  Wien's  Law  of  Radia- 
tion (6). 

§  4.  The  Quantum  Hypothesis.     Planck's  Law  of  Radiation 

Lummer  and  Pringsheim,  however,  refused  to  surrender. 
In  a  fresh  investigation  21  in  1900  they  showed  that  in  the 
region  of  long  waves  Wien's  radiation  formula  undoubtedly  did 
not  agree  with  the  results  of  observation.  As  a  result  of  this, 
Planck,  in  an  important  paper  22  which  must  be  regarded  as 
marking  the  creation  of  the  quantum  theory,  decided  to 
modify  his  method  of  deducing  the  law  of  radiation,  namely, 
by  altering  the  expression  (8)  which  gives  the  mean  energy 
of  the  oscillator,  but  which  is  not  unique.  He  proceeded  as 
follows.23  In  order  to  distribute  the  whole  available  energy 
among  the  oscillators,  he  imagined  this  energy  divided  into 
a  discrete  number  of  finite  "  elements  of  energy "  (energy 
quanta)  of  magnitude  e,  and  supposed  these  energy  quanta 
to  be  distributed  at  random  among  the  individual  oscillators 
exactly  as  a  given  number  of  balls,  say  5,  may  be  distributed 
among  a  certain  number  of  boxes,  say  3.  Each  such  distri- 
bution (of  5  balls  among  3  boxes)  may  obviously  be  carried 


8  THE  QUANTUM  THEORY 

out  in  a  number  of  different  ways,  whereby,  however,  we  are 
not  concerned  with  which  particular  balls  lie  in  which  par- 
ticular boxes,  but  with  the  number  contained  in  each.2*  Now 
since  each  such  "  distribution  "  corresponds  to  a  definite  state 
of  the  system,  it  follows  from  what  has  just  been  said  that 
each  condition  may  be  realised  in  a  number  of  different 
ways,  that  is,  each  condition  is  characterised  by  a  certain 
number  of  possibilities  of  realisation.  This  number  is  called 
by  Planck  the  thermodynamic  probability  W  of  the  condition 
in  question.  For  it  is  obvious  that  the  probability  of  a  con- 
dition or  state  is  the  greater,  i.e.  it  will  occur  the  more  fre- 
quently, the  greater  the  number  of  ways  in  which  it  may 
be  realised.  By  means  of  the  usual  formulae  of  permuta- 
tions and  combinations,  of  which  the  latter  alone  come  into 
consideration  here,  it  was  possible  to  calculate  the  probability 
of  any  given  distribution  of  the  elements  of  energy  among 
the  oscillators,  and  thus  also  the  probability  of  a  given 
energetic  condition  of  the  system  of  oscillators  as  a  func- 
tion of  the  mean  energy  U  of  an  oscillator  and  of  the  energy 
quantum.  Now,  L.  BoUzmann®  has  given  an  extremely 
fertile  rule,  which  connects  the  probability  of  state  W  of  a 
system  with  its  entropy  S,  a  magnitude  which,  as  is  well 
known,  plays  a  similar  role  in  the  second  law  of  thermodyna- 
mics to  that  played  by  energy  in  the  first.  Thus  S  was  ob- 
tained as  a  function  of  U  and  c.  If  now,  on  the  other  hand, 
one  applied  the  second  law  itself,  which  expresses  the  en- 
tropy S  as  a  function  of  the  mean  energy  U  and  the  absolute 
temperature  T,  the  following  result  was  obtained  by  this  cir- 
cuitous process  :  the  entropy,  as  an  auxiliary  magnitude,  was 
eliminated,  and  a  relation  between  U,  T,  and  e  was  gained. 
This  fundamental  result,  first  obtained  by  Planck,  is  as 
follows  :  — 

U  =  —  _!  -  (k  being  a  constant)  .        .     (9) 


But  from  (7)  and  TPtenVDisplacement  Law  (4)  it  follows 
that  for  the  mean  energy  U  of  an  oscillator,  a  relationship  of 
the  following  form  exists  :  — 


THE  QUANTUM  HYPOTHESIS  9 

A  comparison  of  (9)  and  (10)  shows  that  U  assumes  the 
form  required  by  (10)  only  when  e  is  set  proportional  to  v,  the 
frequency.  This  is  an  essential  point  of  Planck's  Theory :  if 
we  are  to  remain  in  agreement  with  Wien's  Displacement  Law, 
the  energy  element  c  must  be  set  equal  to  hv 

t  =  hv         .         .         .         .     (11) 

The  constant  h,  which,  on  account  of  its  dimensions  (energy 
x  time),  is  called  Planck's  Quantum  of  Action,  has  played, 
as  we  shall  see,  a  r61e  of  undreamed-of  importance  in  the 
further  development  of  the  quantum  theory. 

By  combining  the  formulae  (7),  (9),  and  (11)  the  renowned 
radiation  law  of  Planck  follows  at  once  : — 


•        •        -     (12) 
-  1 

which  Planck  first  deduced  in  the  year  1900  in  the  manner 
above  described,  that  is,  by  the  hypothesis  of  energy  quanta. 
In  the  same  year  as  well  as  in  the  following  year  this  Law  of 
Kadiation  was  confirmed  very  satisfactorily  by  H.  Rubens  and 
F.  Kurlbaum  2°  for  long  waves,  and  by  F.  Paschen  Z1  for  short 
waves.  The  later  measurements  of  radiation  emitted  by 
black  bodies,28  particularly  the  exact  work  carried  out  by  E, 
Warburg  and  his  collaborators  at  the  Eeichsanstalt,  have  also 
demonstrated  the  validity  of  Planck's  formula.  In  opposition 
to  this,  W.  Nernst  and  Th.  Wulf&  as  the  result  of  a  critical 
review  of  the  whole  experimental  material  available  up  to  that 
date,  have  recently  shown  the  existence  of  deviations  (up  to 
7  per  cent)  between  the  measured  and  the  calculated  values 
according  to  Planck's  formula,  and  hence  feel  themselves 
constrained  to  decide  against  the  exact  validity  of  Planck's 
formula.  Whatever  view  is  taken  of  this  criticism,  it  is  at  any 
rate  a  powerful  incentive  to  take  up  anew  the  measurement 
of  the  radiation  emitted  by  black  bodies  with  all  the  finesse 
and  precautions  of  modern  experimental  science,  and  thereby 
to  decide  finally  the  important  question  whether  Planck's 
Law  is  exactly  valid  or  not. 

For  short  wave-lengths,  i.e.  high  frequencies  (more  exactly, 


10  THE  QUANTUM  THEORY 

for  high  values  of  =-^\,  Planck's  formula  assumes  the  form 

and  thus  passes  over  into  Wien's  Law  (cf.  formula  (6), 
which,  as  we  have  seen,  was  confirmed  by  experiment  for 
these  frequencies).  In  the  other  limiting  case,  i.e.  for  long 

waves,  low  frequencies  (more  exactly  for  small  values  of  ?-=, 
Planck's  formula  assumes  the  form 

*.-£«•    ....    (H) 

as  is  easily  found  by  developing  the  exponential  function 

efk  as  a  series.  This  limiting  law,  which  has  been  confirmed 
in  the  region  of  long  wave-lengths,  had  been  given  pre- 
viously by  Lord  Rayleigh.30  Planck's  formula  thus  contains 
Wien's  Law  and  Eayleigh's  Law  as  limiting  cases. 

If  we  use  the  wave-length  X  instead  of  the  frequency  v, 
Planck's  Law  takes  the  form 

.         .         .     (15) 

To  make  this  clear,  the  intensity  of  radiation  E^  is  plotted 
in  Pig.  1  as  a  function  of  A.  for  various  values  of  T.  The 
curves  which  exhibit  K.v  as  a  function  of  v  have  a  quite 
similar  appearance.  The  maximum  of  the  ^-curves  lies  at 

the  point  at  which  _A  has  the  value  4-9651. 
It  follows  that 

Xinax  '  T  =  TaSr-T  =  6>042  x  1°9  •  ?  =  8    •     (16) 

4'yoOl  .  K  K 

a  relation,  which  is  identical  in  form  with  Wien's  Displace- 
ment Law  (5a). 

For  the  total  radiation  we  get  from  (12)  or  (15) 


CONSEQUENCES  OF  PLANCK'S  THEORY     11 

an  equation  which  gives  expression  to  the  Stefan- Boltzmann 

Law  31  (3). 

From  (16)  and  (17)  we  recognise  that  the  measurement  (a) 
of  the  total  radiation  (K)  and  (6)  of  the 
wave-length  of  the  maximum  (A.max),  at 
a  fixed  known  temperature,  allows  us  to 
calculate  the  two  constants  h  and  k  of  the 
radiation  formula.32  From  Kurlbaum's 
measurements  of  the  Stefan-Boltzmann 
constant  y,  which  were  available  at  that 
time,  and  from  the  constant  8  of  Wien's 
Displacement  Law  (measured  by  Lummer 
and  Pringsheim)  Planck  33  found  the  fol- 
lowing values : 


h  =  6-548  x  10-27[erg.sec.] 

k  =  1-346.  10  -"fl^l        • 
Ldeg.J 


(18) 


Corresponding    to    the    varying    values 
which  have  been  found  in  the  course  of 
time  for  the  constants  y  and  8,  the  values 
h  and  k  have  undergone  changes  which 
are  not  worth  while 
recording       here. 
For      particularly 

_/l   the  measurement  of 

pIG    i_  the  total  radiation 

— as  we  see  from 

the  strongly  varying  values  given  in  note  15 — has  not  yet 
reached  a  sufficient  degree  of  certainty,  to  allow  a  very  ac- 
curate calculation  of  the  two  radiation  constants  h  and  k  to  be 
based  on  the  Stefan-Boltzmann  constant.  Methods  which 
allow  h  to  be  determined  with  undoubtedly  much  greater 
accuracy  will  be  described  later. 

§  5.  Consequences  of  Planck's  Theory 

The  deduction  of  the  radiation  formula  and  the  determina- 
tion of  its  constants  did  not,  however,  exhaust  the  successes 
of  Planck's  new  theory  ;  on  the  contrary,  important  relation- 
ships of  this  theory  to  other  departments  of  physics  became 


12  THE  QUANTUM  THEORY 

immediately  revealed.  For  it  was  found  3*  that  the  constant 
k  of  the  radiation  formula  is  nothing  other  than  the  quotient 
of  the  absolute  gas  constant  B  (which  appears  in  the  equa- 
tion of  state  of  an  ideal  gas)  and  the  so-called  Avogadro 
number  N,  i.e.  the  number  of  molecules  in  a  grammolecule. 

*-*       '.        .        .        .     (19) 

As  the  value  of  R  is  sufficiently  accurately  known  from 
thermodynamics 


Planck,**  by  making  use  of  the  radiation  measurements,  was 
able  to  calculate  the  value  of  N.  By  using  (18)  he  found 

N  =  6-175  x  1023         .    \.        .     (20) 

The  agreement  of  this  value  with  the  values  deduced  by 
quite  different  methods  is  very  striking.86  Avogadro  s  Law 
forms  the  bridge  to  the  electron  theory.  For  it  is  known 
that  the  electric  charge  which  travels  in  electrolysis  with 
1  gramme-ion,  that  is,  with  N-ions,  is  a  fundamental  con- 
stant of  nature,  which  is  called  the  Faraday.  Its  value  was, 
according  to  the  position  of  measurements  at  that  time, 
9658  .  3  .  1010  electrostatic  units  (the  value  nowadays  ac- 
cepted 37  is  9649-4  .  2-999  .  1010).  If  now  each  monovalent- 
ion  carries  the  charge  e,  of  the  electron,  the  equation 

Ne  =  9658  .  3  .  1010        .         .         .     (21) 
must  hold.     From  this,  by  using  (20),  we  get 

e  =  4-69  x  10-10      electrostatic  units         ..    (22) 

The  value  of  the  electron  charge  thus  calculated  by  Planck 
from  the  theory  of  radiation  differs  only  by  about  2  per  cent 
from  the  latest  and  most  exact  measurements  of  R.  A. 
Millikan**  who  found  the  value 

e  =  4-774  .  10  -  10     electrostatic  units.         .     (23) 
A  truly  astonishing  result. 


CHAPTEE  II 
The  Failure  of  Classical  Statistics 

§  I.  The  Equipartition  Law  and  Rayleigh's  Law  of  Radiation 

IF  these  great  successes  had  justified  faith  in  Planck's 
Theory,  it  was  also  soon  recognised — as  had  already  been 
emphasised  by  Planck  in  his  first  papers — that  the  central 
point  of  the  theory  lay  in  the  Quantum  Hypothesis,  i.e.  in  the 
novel  and  repulsive  conception,  that  the  energy  of  the  oscilla- 
tors of  natural  period  v  was  not  a  continuously  variable 
magnitude,  but  always  an  integral  multiple  of  the  element  of 
energy,  that  is  e  =  hv.  The  recognition  of  the  necessity  of 
this  hypothesis  has  forced  itself  upon  us  more  and  more  in  the 
course  of  time,  and  has  become  established,  more  especially 
through  indirect  evidence,  inasmuch  as  every  attempt  to  work 
with  the  classical  theory  has  led  logically  to  a  false  law  of 
radiation.  For  when  Planck  turned  the  radiation  problem 
into  a  problem  of  probability — for  a  definite  amount  of  energy 
was  to  be  divided  among  the  oscillators  according  to  chance, 
and  the  mean  value  U  of  the  energy  of  an  oscillator  was  to 
be  calculated — it  became  possible  to  apply  the  methods  of 
the  statistical  mechanics  founded  by  Clerk  Maxwell,  L. 
Boltzmann,  and  Willard  Gibbs.  And  the  application  of  these 
methods  to  the  case  in  question  appeared  to  be  demanded 
from  the  start,  if  the  standpoint,  self-evident  in  classical 
physics,  that  the  energy  of  the  oscillator  could  assume  in 
continuous  sequence  all  values  between  0  and  CD  were 
adopted.  What,  then,  did  statistical  mechanics  require? 
One  of  its  chief  laws  is  the  law  of  the  equipartition  of  kinetic 
energy, *  according  to  which  in  a  state  of  statistical  equilibrium 
at  absolute  temperature  T  every  degree  of  freedom  of  a  mechan- 
ical system,  however  complicated,  possesses  the  mean  kinetic 
13 


14  THE  QUANTUM  THEORY 

energy  ^TcT.  In  this  expression  the  constant  k  is  defined  by 
(19),  and  is  thus  the  same  constant  as  that  which  appears 
in  the  Law  of  Eadiation.  A  system  of/  degrees  of  freedom, 
therefore,  possesses  at  a  temperature  Ta  mean  kinetic  energy 
/ .  $kT.  For  example,  the  atom  of  a  monatomic  gas  is  a 
configuration  which  possesses  three  degrees  of  freedom,  if  we 
regard  it  from  the  point  of  view  of  mechanics  as  a  mass- 
point.  Its  kinetic  energy  at  the  temperature  T  has  therefore 
a  mean  value*0  %kT,  independent  of  its  mass,  a  result  which 
has  been  known  in  the  kinetic  theory  of  gases  since  the  time 
of  Maxwell,  and  which  is  deduced  as  a  consequence  of  his 
law  of  distribution  of  velocities. 

Planck's  linear  oscillator,  which  is  essentially  identical 
with  an  electron  vibrating  in  a  straight  line,  possesses  one 
degree  of  freedom ;  its  kinetic  energy  at  the  temperature  T 
has  therefore  the  mean  value  -^kT.  Now  the  mean  potential 
energy  of  the  oscillator  is  equal  to  its  mean  kinetic  energy.*1 
As  a  result,  its  mean  total  energy  (kinetic  plus  potential)  has 
the  value 

C7=  kT       .        .        .        .     (24) 

This  result  of  classical  statistics,  when  combined  with  the 
relation  (7)  deduced  from  classical  electrodynamics,  gives 
Rayleigh's  Law  of  Radiation 

Kv  =  ^kT    .        .        ...     (25) 

which,  as  we  saw  (cf.  (14)),  is  contained  in  Planck's  Law  of 
Radiation  as  a  limiting  case  for  small  values  of  pL  that  is, 

for  long  waves  or  high  temperatures. 

This  Law  of  Radiation  of  Bayleigh  which,  deduced  as  it 
is  from  the  fundamental  principles  of  classical  statistics  and 
electrodynamics,  should  be  able  to  claim  general  validity  for 
all  frequencies  and  all  temperatures,  stands  none  the  less  in 
glaring  contradiction  to  observation.  For  while  all  observed 
curves  of  distribution  of  energy  of  a  black  body  (i.e.  Kv  plotted 
as  a  function  of  v,  T  being  constant)  always  show  a  maximum, 
the  curve  expressed  by  (25)  rises  without  limit  for  rising 
values  of  v,  and  therefore  gives  for  the  sum  K  —  2  /  Kvdv  an 
infinitely  large  value. 


FRUITLESS  ATTEMPTS  AT  IMPROVEMENT     15 

§  2.  Fruitless  Attempts  at  Improvement 

From  very  different  quarters  and  in  the  most  varied  ways 
attempts  were  made,  as  time  went  on,  to  escape  from 
Rayleigh's  Law  without  discarding  classical  statistical 
mechanics.  All  in  vain.  Thus  «7.  H.  Jeans,*2  without 
making  use  of  a  "material"  oscillator,  considered  only  the 
radiation  as  such  in  an  enclosure,  and  distributed  the  whole 
energy  of  radiation  according  to  the  Law  of  Equipartition 
over  the  individual  "  degrees  of  freedom  of  radiation  "  (which 
are  here  the  individual  vibrations  that  are  possible  in  an  en- 
closure). Further,  H.  A.  Lorentz**  deduced  in  a  penetrating 
investigation  the  thermal  radiation  of  the  metals,  starting  from 
the  conception  that  the  free  "  conduction  electrons,"  which 
carry  the  current,  produce  the  radiation  by  their  collisions 
with  the  atoms,  and  applying  the  Law  of  Equipartition  to 
the  motion  of  these  electrons.  The  problem  was  attacked 
in  a  somewhat  different  fashion  by  A.  Einstein  and  L.  Hop/.** 
They  imagined  the  Planck  oscillator  firmly  attached  to  a 
molecule,  and  then  considered  this  complex  exposed  to  the 
radiation  and  the  impacts  of  other  molecules.  The  Law  of 
Radiation  could  then  be  deduced  from  the  condition  that  the 
impulse,  which  the  impacts  of  the  molecules  give  to  the  com- 
plex, must  not  on  the  average  be  changed  by  the  impulses, 
which  the  radiation  gives  to  the  oscillator.  We  may  also 
mention  a  paper  of  A.  D.  Fokker**  which  was  supplemented 
by  M.  Planck.*6  In  this,  by  the  aid  of  a  general  law  due  to 
Einstein,  the  statistical  equilibrium  between  the  radiation 
and  a  large  number  of  oscillators  was  examined  on  the  basis 
of  the  classical  theories.  All  these  different  ways  ended, 
however,  at  the  same  point ;  they  all  led  to  Rayleigh's  Law. 
And  finally,  at  the  Solvay  Congress  in  Brussels  in  1911, 
H.  A.  Lorentz «  showed,  in  the  most  general  manner 
imaginable,  that  we  arrive  of  necessity  at  this  wrong  law, 
if  we  assume  the  validity  of  Hamilton's  Principle  and  of 
the  Principle  of  Equipartition  for  the  totality  of  the  pheno- 
mena (of  mechanical  and  electromagnetic  nature)  which 
take  place  in  an  enclosure  containing  radiation,  matter,  and 
electrons.  Only  in  the  limiting  case  of  high  temperatures  or 
small  frequencies  do  the  results  of  the  classical  theory  agree 
with  the  results  of  observation.*8 


CHAPTEE  III 

The  Development  and  the  Ramifications  of  the 
Quantum  Theory 

§  i.  The  Absorption  and  Emission  of  Quanta 

A  S  stated  above,  the  conviction  was  bound  to  establish 
Xl^tself  that  every  attempt  to  deduce  the  laws  of  radiation 
on  the  basis  of  classical  statistics  and  electrodynamics  was 
doomed  from  the  outset  to  failure,  and  it  was  necessary  to 
introduce  a  hitherto  unknown  discontinuity  into  the  theory. 
It  was,  of  course,  clear  that  this  "  atomising  of  energy  "  would 
conflict  sharply  with  existing  and  apparently  well-founded 
theories.  For  if  the  energy  of  the  Planck  oscillator  was  only 
to  amount  to  integral  multiples  of  e  =  hv,  and  therefore  was 
only  to  be  able  to  have  the  values  0,  e,  2e,  3e  .  .  .  then,  since 
the  oscillator  only  changes  its  energy  by  emission  and  ab- 
sorption, the  conclusion  was  inevitable  that  oscillators  cannot 
absorb  and  emit  amounts  of  energy  of  any  magnitude  but  only 
whole  multiples  of  e.  (Quantum  emission  and  quantum 
absorption.}  This  conclusion  is  in  absolute  contradiction  to 
classical  electrodynamics.  For,  according  to  the  electron 
theory,  an  electromagnetic  oscillator,  for  instance  a  vibrat- 
ing electron,  emits  and  absorbs  in  a  field  of  radiation  perfectly 
continuously,  that  is  to  say,  in  sufficiently  short  times  it  emits 
or  absorbs  indefinitely  small  amounts  of  energy. 

§  2.  Einstein's  Light-quanta  ;  Phenomena  of  Fluctuation  in  a  Field 
of  Radiation 

Thus  at  the  very  entrance  into  the  new  country  there 
yawned  a  gulf,  which  had  either,  in  view  of  the  previous 
success  of  the  classical  theory,  to  be  bridged  over  by  a  com- 
promise ;  or,  failing  this,  tradition  would  have  to  be  discarded 
and  the  gap  would  be  relentlessly  enlarged.  Einstein  felt  him- 
16 


EINSTEIN'S  LIGHT-QUANTA  17 

self  compelled  to  take  the  latter  radical  course.  On  the  basis 
of  very  original  considerations,49  he  set  up  the  hypothesis  that 
the  energy  quanta  not  only  played  a  part,  as  Planck  held,  in 
the  interaction  between  radiation  and  matter  (resonators  or 
oscillators),  but  that  radiation,  when  propagated  through  a 
vacuum  or  any  medium,  possesses  a  quantum-like  structure 
(Light- quantum  hypothesis).  Accordingly,  all  radiation  was 
to  consist  of  indivisible  "  radiation  quanta  "  ;  when  the  energy 
is  being  propagated  from  the  exciting  centre,  it  is  not  divided 
evenly  in  the  form  of  spherical  waves  over  ever-increasing 
volumes  of  space,  but  remains  concentrated  in  a  finite  numbe 
of  energy  complexes,  which  move  like  material  structures, 
and  can  only  be  emitted  and  absorbed  as  whole  individuals. 
Einstein  believed  himself  forced  to  this  strange  conception, 
which  breaks  with  all  the  observations  that  appear  to 
support  the  undulatory  theory,  by  several  investigations, 
all  of  which  led  to  the  same  conclusion.  He  was  per- 
suaded to  this  view  by  the  result  of  calculations  dealing 
with  certain  phenomena  of  fluctuation,  phenomena  which 
are  familiar  to  us  in  statistics  and  particularly  in  the  kinetic 
theory  of  gases.  It  is  well  known  that  in  a  gas  which 
contains  n  molecules  in  a  volume  v0,  the  spatial  distribution 
of  these  molecules  is  far  from  constant,  being  subject  to  vari- 
ation on  account  of  the  motion  of  the  molecules.  Indeed,  in 
principle,  extreme  cases  are  possible  as  that,  for  example,  in 
which  all  n  molecules  are  collected  at  a  given  moment  in  a 
fractional  part  v(<v0)  of  the  volume.  The  probability  of 
this  rare  constellation  is  known  to  be 

.         .         .         .     (26) 

an  extraordinarily  small  number  when  n  is  great ;  that  is  to 
say,  the  event  in  question  occurs  extremely  rarely. 

Now,  the  spatial  density  of  the  radiation  enclosed  within  a 
volume  v(}  is  subject  to  quite  analogous  variations.  If  E  is 
the  total  energy  of  the  radiation  (supposed  to  be  monochro- 
matic) and  if  its  frequency  v  is  so  great,  or  its  temperature 
so  low,  that  Wien's  Law  of  Radiation  holds  for  it,  then  the 
probability  that  the  whole  radiation  occupies  the  partial 
volume  -u«  VQ)  is,  according  to  Einstein,*0 


18  THE  QUANTUM  THEORY 

io-^\C  .     (27) 


A  comparison  with  (26)  shows  that  the  radiation,  within 
the  limits  of  validity  of  Wien's  Law,  behaves  as  if  it  were  made 

wp  of  n  (  =  r-  j  independent  complexes  of  energy,  each  of  mag- 
nitude hv. 

Two  other  investigations  si  of  Einstein  led  to  the  same 
conclusion.  In  the  first,  a  very  large  volume  filled  with 
black-body  radiation  is  considered,  which  communicates  with 
a  small  volume  v.  If  E  is  the  momentary  energy  of  the 
radiation  of  frequency  v  in  the  volume  v,  this  energy  varies, 
as  is  known,  irregularly  with  the  time  about  a  mean  value 
E ;  the  magnitude  e  =  E  -  E  is  called  the  fluctuation  of  the 
energy.  Now,  the  general  theory  of  statistics  leads  to  the 
following  value  fl2  for  the  mean  square,  that  is,  for  e2, 

P-fcT'.g;.        .        .        .-  (28) 

If  we  replace  E  by  the  value  obtained  from  Planck's  Law  of 
Eadiation,  we  obtain  for  the  mean  square  of  fluctuation  an 
expression  with  two  terms,93  in  which  only  one  term  can  be 
calculated  on  the  basis  of  the  classical  undulatory  theory; 
the  second,  which  greatly  exceeds  the  first  in  magnitude 
when  the  density  of  radiant  energy  is  low  (that  is,  at  high 
frequencies  or  at  low  temperatures,  in  short,  when  Wien's 
Law  is  valid),  can  only  be  understood  when  we  again  picture 
the  radiation  as  composed  of  indivisible  energy-quanta. 

The  second  of  Einstein's  two  investigations,  to  which  we 
referred  above,  deals  with  the  fluctuations  of  impulse  which 
a  freely  movable  reflecting  plate  is  subjected  to  in  a  field  of 
black-body  radiation  on  account  of  the  irregular  fluctuations 
of  the  pressure  of  radiation.  If,  in  addition,  the  plate  is  sub- 
jected to  the  irregular  blows  of  gas-molecules,  under  the 
influence  of  which  it  executes  Brownian  movements,  there 
must  be  equilibrium  between  the  impulses  which  the  mole- 
cules on  the  one  hand,  and  the  radiation  on  the  other,  im- 
part to  the  plate.  If,  now,  we  assume  Planck's  Law  to  hold 
for  the  radiation,  there  again  follows  for  the  mean  square  of 
the  variations  in  impulse  due  to  the  radiation  an  expression 


TRANSFORMATION  OF  LIGHT-QUANTA     19 

in  two  terms,  only  one  of  which  is  explained  by  the  un- 
dulatory  theory  of  light.  The  other  term  points  to  a 
quantum-like  structure  of  the  radiation,  and  this  suggests  the 
introduction  of  the  light-quantum  hypothesis. 

§  3.  Transformation  of  Light-quanta  into  other  Light-quanta  or 
Electronic  Energy 

However  strange  this  hypothesis  appeared,  it  was  not  to  be 
denied  that  it  was  capable  of  explaining  simply  and  naturally 
a  number  of  phenomena  which  completely  baffled  the  un- 
dulatory  theory.  A  very  striking  example  of  this  is  afforded 
by  the  laws  of  phosphorescence,  investigated  by  P.  Lenard 
and  his  co-workers,  and  especially  by  Stokes'  Law.  For  if 
vp  is  the  frequency  of  the  phosphorescent  light  emitted, 
and  ve  the  frequency  of  the  light  exciting  phosphorescence, 
then,  according  to  Einstein's  conception,34  one  quantum  hve 
of  the  exciting  radiation  is  changed  through  absorption  by 
the  atom  of  the  phosphorescent  substance  into  one  quantum 
hvp  of  the  light  of  phosphorescence.  According  to  the  prin- 
ciple of  energy,  we  must  have  hve  !>  hvp,  i.e.  ve  >  vp.  And 
this  is  Stokes'  Law. 

Further,  another  fact  in  the  realm  of  phosphorescence 
phenomena  speaks  against  the  undulation  hypothesis  and  in 
favour  of  that  of  light-quanta.  According  to  the  classical 
undulatory  theory,  all  molecules  of  a  phosphorescent  body 
on  which  a  light- wave  impinges,  should  absorb  energy  from 
the  wave,  and  thus  all  simultaneously  become  able  to  emit 
phosphorescent  light.  In  reality,  relatively  only  very  few 
molecules  are  excited  to  phosphorescence  at  the  same  time, 
and  only  gradually,  in  the  course  of  time,  does  the  number  of 
molecules  excited  increase.  It  would  thus  appear  as  if  the 
light-wave  falling  on  the  phosphorescent  body  has  not  equal 
intensity  along  its  whole  front — as  the  classical  theory 
assumes — but  rather  as  if  it  consists  of  single  energy-com- 
plexes thrown  out  by  the  source  of  light,  so  that  the  wave-point 
possesses,  as  it  were,  a  "beady"  structure,  in  which  active 
portions  (light-quanta)  alternate  with  inactive  gaps. 

This  conception  of  the  "  beady  "  wave-front  had  played  a 
part  before  the  advent  of  Einstein's  hypothesis  of  light-quanta. 
J".  /.  Thomson 8B  had  tried  to  make  use  of  it  to  explain  the 


20  THE  QUANTUM  THEORY 

fact  that,  when  a  gas  is  ionised  by  ultra-violet  light  or  Kontgen 
rays,  only  a  relatively  extremely  small  number  of  gas-mole- 
cules are  ionised.  This  is  a  phenomenon  which  is  quite 
analogous  to  the  above-named  phenomenon  of  phosphor- 
escence ;  for  these,  too,  according  to  Lenard's  view,  the  exci- 
tation consists  in  the  disjunction,  through  the  agency  of  the 
radiation,  of  electrons  from  the  molecules  of  the  phosphor- 
escent body,  and  these  electrons  attach  themselves  to  "  storage 
atoms."  On  the  return  of  these  electrons  to  the  parent 
molecules,  energy  is  set  free  and  sent  out  as  phosphorescent 
light.  The  ionisation  of  gases  by  ultra-violet  light  or  Ront- 
gen  rays  M  is  also  capable  of  being  explained  naturally  by  the 
light-quantum  hypothesis.  If  we  suppose  with  Einstein, 
that  one  light-quantum  hv  is  used  up  in  ionising  one  mole- 
cule, then  hv  >  /,  where  /  is  the  work  required  to  ionise 
one  molecule,  that  is  to  say,  to  remove  an  electron  from  it. 
We  have  under  consideration  here  a  phenomenon  which  be- 
longs to  the  great  branch  of  photo-electric  phenomena,*1  i.e. 
the  liberation  of  electrons  from  gases,  metals,  and  other  sub- 
stances by  the  action  of  light.  According  to  the  hypothesis 
of  light-quanta,  in  all  these  processes  light-quanta  are  changed 
into  kinetic  energy  of  the  electrons  hurled  off  from  the  body. 
If  we  again  adopt  Einstein 's  standpoint,  according  to  which 
one  light-quantum  hv  is  transformed  into  the  kinetic  energy 
of  one  projected  electron,  we  must  have  the  following  re- 
lation M  for  the  energy  of  emission  of  the  emitted  electrons, 
each  having  a  mass  ra  : 

fynV2  =  hv  -  P  .         .        .         .     (29) 

This  is  called  Einstein's  Law  of  the  Photo-electric  Effect.  In 
this,  P  is  the  work  that  has  to  be  done  to  tear  the  electron 
away  from  the  atom,  and  to  project  it  from  the  point  at 
which  it  is  torn  from  the  atom  up  to  the  point  at  which  it 
leaves  the  surface  of  the  body.  For  the  energy  of  the  emitted 
electrons  we  thus  obtain  a  linear  increase  with  the  periodicity 
of  the  light  which  releases  them.  This  law,  which  many  in- 
vestigators have  attempted  to  prove,  with  varying  success, 
has  recently  been  verified  by  B.  A.  Millikan*9  for  the  normal 
photo-electric  effect w  of  the  metals  Na  and  Li  with  such  a 
degree  of  accuracy  that  we  can  actually  use  this  method  for 


ELECTRONIC  ENERGY  21 

the  exact  determination  of  h.  The  value  found  by  Millikan, 
h  =  6-57  x  10 ~27,  is  in  good  agreement  with  the  value 
h  =  6'548  x  10'27  found  by  Planck  from  radiation  measure- 
ments. 

In  an  entirely  similar  manner  as  was  used  for  the 
phenomena  of  phosphorescence,  the  phenomena  of  fluores- 
cence in  the  regions  of  the  Rontgen  and  visible  radiations  may 
be  explained  by  the  hypothesis  of  light-quanta.  The  in- 
vestigations of  Ch.  Barkla,  Sadler,  M.  de  Broglie,  and 
E.  Wagner 61  have  shown  the  following :  if  a  body  is  inun- 
dated with  Eontgen  rays,  and  if  the  absorption  of  these  rays  by 
the  body  is  measured  whilst  the  hardness  (i.e.  the  frequency 
ve)  of  the  rays  is  varied,  the  absorption,  as  we  pass  from 
lower  to  higher  ve,  suddenly  increases  to  a  high  value  for  a 
certain  value  of  ve.  At  the  same  moment  the  body  begins, 
at  the  expense  of  the  energy  absorbed,  to  emit  a  secondary 
Rontgen  radiation  characteristic  of  the  body  itself  in  the  form 
of  a  line  spectrum.  It  further  appears  that  all  lines  emitted 
have  a  lower  v  than  that  of  the  exciting  radiation.  As  a 
matter  of  fact,  the  hypothesis  of  light-quanta  requires  that  the 
radiation-quantum  hv  of  all  rays  emitted  as  secondary  radia- 
tion should  be  smaller  than  the  quantum  hv  of  the  primary 
exciting  rays.  For  example,  the  region  of  frequencies  which 
serves  to  excite  the  "  ^-series "  stretches  from  a  sharply 
defined  limit  vk  (the  so  called  "  edge  of  the  absorption  band  ") 
upwards  towards  higher  frequencies;  whereby  vk  is  some- 
what larger  than  the  hardest  known  line  (y)  of  the  ^-series. 
In  other  words,  the  excitation  of  secondary  Rontgen  radiation 
by  primary  Rontgen  rays  also  obeys  Stokes'  Law. 

§  4.  The  Transformation  of  Electronic  Energy  into  Light-quanta 

It  is  very  significant,  that  the  transformation  of  light- 
quanta  into  kinetic  enei'gy  of  electrons  is  also,  as  it  were, 
"reversible,"  that  is,  the  opposite  process  also  occurs  in 
nature,  by  which  light-quanta  result  from  the  kinetic  energy 
of  charged  particles.  A  good  example  of  processes  of  this 
kind  is  afforded  by  the  generation  of  Rontgen  rays  by  the 
impact  of  quickly-moving  electrons  (cathode  rays)  on  matter. 
If,  say,  the  characteristic  X"-series  of  a  certain  element  is  to 


22  THE  QUANTUM  THEORY 

be  generated  by  the  impact  of  cathode  rays  upon  an  anti- 
cathode  formed  of  the  said  element,  then  the  kinetic  energy 
E  of  an  impinging  electron  must  exceed  a  critical  value  EK. 
For  if  we  imagine  E  changed  into  a  light-quantum  hve,  then 
ve  must  fall  within  the  region  of  excitation  of  the  JT-series,  and 
must  thus  be  ^  VK  (VK  being  the  frequency  of  the  edge  of  the 
absorption  band).  It  follows  that  E  ^>  hvK(  =  EK).  From  this 
there  follows  an  important  relation  between  the  frequency  VK 
of  the  edge  of  the  absorption  band  and  the  critical  value  EK 
of  the  electronic  energy,  i.e.  the  smallest  value  of  the  energy 
at  which  the  electron  is  just  able  to  generate  the  required 
secondary  radiation.  This  quantum-relation  EK  =  hvK  has 
proved  quite  correct  according  to  measurements  carried  out 
by  D.  L.  Webster  &  and  E.  Wagner  w  and  conversely  presents, 
when  EK  and  VK  are  sufficiently  accurately  known,  a  method 
for  the  determination  of  /i.6* 

Now,  it  is  known  that  the  cathode  rays,  on  striking  the 
anti-cathode,  do  not  merely  excite  the  characteristic  Eontgen 
radiation,  that  is  a  line  spectrum,  but  excite  a  continuous 
spectrum  as  well,  the  so-called  "  impulse  radiation  "  (Brems- 
strahlung).  If  we  therefore  select  any  frequency  v  of  this 
continuous  spectrum,  the  ideas  of  the  hypothesis  of  light- 
quanta  immediately  suggest  the  conclusion  that  a  definite 
minimum  energy  Em  of  the  impinging  electrons  is  necessary 
to  excite  this  frequency  v,  and  that  we  must  have  Em  =  hv. 
The  investigations  of  D.  L.  Webster  &  W.  Duane  and  F.  L. 
Hunt**,  A.  W.  Hull  and  M.  Bice^E.  Wagner, M  F.  Dessauer 
and  E.  Back66  have  confirmed  these  formulae  with  the 
greatest  accuracy,  and  thus  form  the  foundation  of  one  of 
the  most  trustworthy  methods  for  the  precise  measurement 
of  the  magnitude  h.  The  following  values  were  obtained : 
h  =  6-50  x  10-27  (Duane-Hunt) ;  h  =  6-53  x  10 -2~  (Webster) ; 
h  =  6-49  x  10  ~27  (Wagner). 

We  also  meet  with  similar  phenomena  in  the  visible  and 
neighbouring  regions  of  the  spectrum.  Thus  /.  Franck 
and  Cr.  Hertz m  showed  that  the  impact  of  electrons  upon 
mercury  vapour  molecules  can  be  used  to  excite  a  definite 
characteristic  fluorescence  line  of  mercury  of  wave-length 
A0  =  25364°  (i.e.  v0  =  1-183  . 1016),  if  the  kinetic  energy  of  the 


HYPOTHESIS  OF   LIGHT-QUANTA  28 

electron  exceeds  a  certain  critical  value  EQ.  In  this  con- 
nexion they  found  that  the  relation  E0  =  hv0  was  again 
fulfilled  with  great  accuracy.70  We  shall  return  to  these 
experiments  and  others  connected  with  them  later,  since  they 
play  an  important  part  in  confirming  the  most  recent  model 
of  the  atom. 

§  5.  Other  Applications  of  the  Hypothesis  of  Light-quanta 

In  a  considerable  number  of  other  cases,  which  shall  only 
be  noticed  shortly  at  this  point,  the  hypothesis  of  light-quanta 
has  proved  of  value,  especially  in  the  hands  of  /.  Stark 71  and 
Einstein.  Thus  Stark  ra  has  made  use  of  this  hypothesis  to 
interpret  the  fact  that  the  canal-ray  particles  emit  their 
"  kinetic  radiation  "  only  when  their  speed  exceeds  a  certain 
value.  He  has  also  propounded  general  laws  for  the  position 
of  band-spectra  of  chemical  compounds  by  arguing  on  the  basis 
of  the  hypothesis  of  light-quanta.78  Finally,  Einstein 7*  and 
Stark 78  have  considered  photo-chemical  reactions  from  the 
standpoint  of  the  hypothesis  of  light-quanta  and  have  enun- 
ciated a  fundamental  law,  which  has  been  verified,  at  least 
partially,  by  the  detailed  investigations  of  E.  Warburg.™ 

§6.  Planck's  Second  Theory 

In  spite  of  all  the  successes  which  the  quantum  hypothesis 
of  light  is  able  to  show,  we  must  not  leave  out  of  consideration 
that  this  radical  view,  at  least  in  its  existing  form,  is  very 
difficult  to  bring  into  agreement  with  the  classical  undulatory 
theory.  Since  on  the  one  hand  the  phenomena  of  interference 
and  diffraction,  in  all  their  observed  minutiae,  are  excellently 
described  by  the  wave-theory,  but  offer  almost  insuperable 
difficulties  to  the  quantum  theory  of  light,  it  is  easy  to  under- 
stand that  few  scientists  could  make  up  their  minds  to  ap- 
prove of  such  a  far-reaching  change  in  the  old  and  well-tested 
conception  of  the  propagation  of  light,  a  change  that  entailed 
perhaps  its  complete  abandonment.  This  more  cautious  and 
conservative  standpoint  was  taken  up  by  M.  Planck,  who 
retains  it  to  this  day,  inasmuch  as  he  preferred  to  locate  the 
quantum  property  in  matter  (the  oscillators) — or  at  least  to 
confine  it  to  the  process  of  interaction  between  matter  and 


24  THE  QUANTUM  THEORY 

radiation — while  endeavouring  to  retain  the  classical  wave- 
theory  for  the  propagation  of  radiation  in  space.  None  the 
less,  serious  hindrances  had  already  intruded  themselves  in 
the  development  of  his  first  quantum  hypothesis  (quantum 
emission  and  quantum  absorption).  For  H.  A.  Lorentz1?1 
pointed  out  quite  rightly  that  the  conception,  especially  of 
quantum  absorption,  leads  to  peculiar  difficulties.  He  showed 
that  the  time  which  an  oscillator  requires  for  the  absorption 
of  a  quantum  of  energy  turns  out  to  belong  to  an  improbable 
degree  when  the  external  field  of  radiation  is  sufficiently  weak. 
Moreover,  it  would  be  possible  to  interrupt  the  radiation  at 
will  before  the  oscillator  had  absorbed  a  whole  quantum.  As 
a  result  of  these  objections  Planck  determined  to  modify  the 
quantum  hypothesis  as  follows.78  Absorption  proceeds  con- 
tinuously and  according  to  the  laws  of  classical  electrodynamics  : 
the  energy  of  the  oscillators  is  therefore  continuously  variable,  and 
can  assume  any  value  between  0  and  oo  .  On  the  other  hand, 
emission  occurs  in  quanta,  and  the  oscillator  can  emit  only 
when  its  energy  amounts  to  just  a  whole  multiple  of  €  =  hv. 
Whether  it  then  emits  or  not  is  determined  by  a  law  of  prob- 
ability. But  if  it  does  emit,  then  it  loses  its  whole  momentary 
energy,  and  therefore  emits  quanta.  Between  two  emissions  its 
energy -content  grows  by  absorption  continuously  and  in  pro- 
portion to  the  time. 

According  to  this  second  theory  of  Planck,  which  is  called 
the  theory  of  quantum  emission,  the  mean  energy  U  of  a 

linear  oscillator  is  ~  greater  than  in  the  first  theory.79    While 

in  the  former  case  the  mean  energy  of  the  oscillator  at  abso- 
lute zero  was  equal  to  zero  (see  equation  (9)  from  which, 
when  T  =  0,  U  —  0),  in  the  case  of  this  second  theory  it  is 

equal  to  — .  The  oscillators  retain  therefore  at  the  zero- 
point  a  zero-point  energy  of  value  -£  as  a  mean,  inasmuch 

2 

as  they  assume,  when  T  =  0,  all  possible  energies  between 
0  and  hv.  Nevertheless,  this  theory  also,  when  the  relation 
(7)  is  correspondingly  modified,  leads  to  Planck's  Law  of 
Kadiation. 

In  the  course  of  time  Planck  has  made  several  further 


ZERO-POINT  ENERGY  25 

attempts  80  to  enlarge  and  modify  this  second  theory  too.  For 
example,  he  has  temporarily  assumed  the  emission  also  to  be 
continuous,  and  relegated  the  quantum  element  to  the  excita- 
tion of  the  oscillators  by  molecular  or  electronic  impacts.  He 
has,  however,  repeatedly  returned  in  essentials  to  the  second 
form  of  his  theory  (continuous  absorption,  quantum  emission). 

§  7.   Zero-point  Energy 

In  more  than  one  direction,  this  theory  has  had  further 
results.  The  appearance  of  the  mean  zero-point  energy, 
which  is  peculiar  to  this  second  theory  of  Planck,  became 
the  starting-point  of  a  series  of  researches,  in  which  certain 
physicists,  going  beyond  Planck,  postulated  the  existence  of 
a  true  (not  mean]  zero-point  energy  equal  for  all  oscillators. 
On  this  basis,  Einstein  and  0.  Stern  81  have  given  a  deduction 
of  Planck's  Law  which  avoids  all  discontinuities  other  than 
the  existence  of  this  zero-point  energy. 

In  the  year  1916,  Nernst 82  took  a  still  more  radical  step  in 
postulating  the  existence  of  a  "  zero-point  radiation  "  which 
was  also  to  be  present  at  the  absolute  zero  of  temperature 
and  was  to  exist  independently  of  heat  radiation,  filling  the 
whole  of  space,  and  such  that  the  oscillators,  as  well  as  all 
molecular  structures,  set  themselves  in  equilibrium  with  it  by 
taking  up  the  zero-point  energy.  Even  if  we  regard  these 
views  more  or  less  sceptically,  one  thing  cannot  be  ignored  : 
many  facts  undoubtedly  support  the  conception  that  at  the 
absolute  zero  by  no  means  all  motion  has  ceased.  We  need 
only  draw  attention  to  the  fact,  that,  according  to  the  view  of 
F.  Richarz*z  P.  Langevin&  and  according  to  the  experiments 
of  Einstein,  W.  J.  de  Haas *»  and  E.  Beck**  Para-  and  Dia- 
magnetism  are  produced  by  rotating  electrons  and  that  this 
magnetism  remains  in  existence  down  to  the  lowest  tempera- 
tures. 

§  8.   Theory  of  the  Quantum  of  Action 

In  yet  another  respect  has  Planck's  theory  proved  stimu- 
lating, in  virtue  of  a  special  formulation  which  Planck  gave 
it87  at  the  Solvay  Congress  in  Brussels  during  1911.  For 
here  Planck  gave  expression  for  the  first  time  to  the  idea 
that  the  appearance  of  energy-quanta  is  only  a  secondary 


26 


THE  QUANTUM  THEORY 


matter,  being  only  the  consequence  of  a  deeper  and  more 
general  law.  This  law,  which  is  to  be  regarded  as  the  pre- 
cursor of  the  latest  development  of  the  doctrine  of  quanta, 
may  be  formulated  as  follows  :  Suppose  the  momentary  state 
of  a  Planck  oscillator,  say  a  linearly  vibrating  electron,  to  be 
defined  according  to  Gibb's  method  by  its  displacement  q  from 
its  position  of  rest  and  by  its  impulse  or  momentum  p,  and 
suppose  it  to  be  represented  in  a  q-p  plane  (the  state-  or 
phase-plane).  Every  point  of  the  q-p  plane,  that  is,  every 
phase-point,  corresponds  to  a  definite  momentary  condition  of 
the  oscillator.  The  postulate  is  then  made  that  not  all  points 
of  this  plane  of  states  are  equivalent.  On  the  contrary,  there 


FIG.  2. 

are  certain  states  of  the  oscillator  which  are  distinguished  by 
a  peculiarity.  The  totality  of  the  phase-points  that  cor- 
respond to  these  peculiar  states  form  a  family  of  discrete 
curves  which  surround  one  another.  In  the  case  of  the 
Planck  oscillator  these  curves  are  concentric  ellipses  (see 
Fig.  2)  which  divide  the  phase-plane  into  ring-like  strips. 
The  postulate  of  the  quantum  theory  now  consists  in  this, 
that  these  ring  strips  all  possess  the  same  area  h.  If  we 
calculate  on  this  basis  the  energy  possessed  by  an  oscillator 
in  one  of  these  unique  states,  we  find80  that  it  is  a  whole 
multiple  of  hv.  These  special  states  (represented  in  the 
phase-plane  by  the  points  of  the  discrete  ellipses)  are,  there- 


THEORY  OF  THE  QUANTUM  OF  ACTION     27 

fore,  according  to  Planck's  first  theory,  the  only  dynamically 
possible  and  stable  states  of  the  oscillator.  If  an  oscillator 
emits  or  absorbs,  its  phase-point  jumps  from  one  ellipse  to 
another.  The  state  of  affairs  is  different  if  we  accept  Planck's 
second  theory.  According  to  this,  all  conditions  of  the  oscil- 
lator, that  is  all  points  on  the  phase-plane,  are  dynamically 
possible.  On  the  other  hand,  emission  takes  place  only  in 
the  states  specially  distinguished  by  the  ellipses.  Seen  from 
this  new  point  of  view,  the  energy-quanta  are,  therefore,  only 
a  result  of  the  partitioning  of  the  phase-plane.  Mathe- 
matically, we  may  express  this  "  structure  of  the  phase-plane  " 
thus  :  the  ?ith  unique  curve  encloses  a  surface  of  area  nh,  or, 
in  symbolic  language, 


I"  [dqdp  =   \ 


nh       „         .         '.     (30) 


The  double  integral  is  taken  over  the  surface  ;  the  single 
integral  is  taken  around  the  boundary  curve  of  the  nth 
ellipse. 

On  this  basis  for  systems  of  one  degree  of  freedom,  which 
is  called  Planck's  theory  of  "  the  action-quantum  "  —  for  h  has 
the  dimensions  of  an  action  —  the  modern  extension  of  the 
quantum  theory  for  several  degrees  of  freedom  has,  as  we 
shall  see,  been  erected. 

Further,  a  line  of  argument  proposed  and  developed  by 
A.  Sommerfeld  takes  its  origin  here.  Starting  from  the  fact 
just  mentioned,  that  Planck's  constant  h  possesses  the  dimen- 
sions of  action  (energy-time),  Sommerfeld  set  up  the  hypo- 
thesis w  that  for  every  purely  molecular  process,  say  the  release 
of  an  electron  in  the  photo-electric  effect,  or  the  stopping  of 
an  electron  by  the  anticathode  in  the  generation  of  Eontgen 

rays,  the  quantity  called  action  (L  -  V)dt,  known  to  us 
V  J° 

from  Hamilton's  Principle,  has  the  value  0-  .     Here  L  and  V 

Air 

are  the  kinetic  and  potential  energies  of  the  electron  respec- 
tively, T  is  the  duration  of  the  molecular  process4!  say,  for 
example,  the  time  which  is  required  for  the  release  of  the 
electron  from  the  atomic  complex  during  the  photo-electric 
effect,  or  the  stopping  of  the  electron  by  the  anti-cathode. 


28  THE  QUANTUM  THEORY 

This  formulation  of  the  quantum  hypothesis  is,  as  it  were, 
an  expression  of  the  well-known  fact  that  large  amounts  of 
energy  are  absorbed  or  given  up  in  short  times,  whereas  small 
amounts  are  absorbed  or  emitted  in  longer  times  by  the 
molecules,  so  that  on  the  whole  the  product  of  the  energy 
transferred  and  the  duration  of  the  time  of  exchange  is  a 
constant.  In  fact,  fast  cathode  rays,  for  example,  are  stopped 
by  matter  in  a  shorter  time — and  therefore  generate  harder 
Kontgen  rays — than  slow  cathode  rays.  Sommerfeld  has 
applied  his  theory  successfully  to  the  mechanism  of  the 
generation  of  Rontgen  rays  and  y-rays.90  Sommerfeld  and 
P.  Debye  w  have  worked  out  on  the  same  basis  a  theory  of 
the  photo-electric  effect,  which,  like  the  hypothesis  of  light- 
quanta,  also  leads  to  Einstein's  Law  (29). 


CHAPTEK  IV 

The  Extension  of  the  Doctrine  of  Quanta  to  the 
Molecular  Theory  of  Solid  Bodies92 

§  i.  Dulong  and  Petit's  Law 

IT  was  a  particularly  fortunate  circumstance  for  the  con- 
solidation of  the  doctrine  of  quanta  that  the  failure  of 
classical  statistics  was  not  confined  to  the  theory  of  radiation, 
but,  as  appears  later,  extended  to  the  molecular  theory  of  solid 
bodies.  Thus  there  arose  in  quite  another  field  a  strong  sup- 
port for  the  quantum  hypothesis,  namely,  in  the  field  of  Atomic 
Heats.  The  Atomic  Heat  of  a  substance  (in  the  case  of  poly- 
atomic bodies  we  say  the  "  Molecular  Heat  ")  is  defined  as  the 
product  of  its  specific  heat  and  its  atomic  weight  (or  molec- 
ular weight) ;  or,  otherwise  expressed,  it  is  that  amount  of 
heat  which  must  be  communicated  to  a  "  gramme-atom  "  m 
(or  gramme-molecule)  of  the  body,  in  order  that  its  tempera- 
ture may  be  raised  by  one  degree.  According  to  our  present 
conceptions,  the  thermal  content  of  a  monatomic  solid,  say 
a  crystal,  is  nothing  more  than  the  energy  of  the  elastic 
vibrations  of  its  atoms,  which  are  arranged  in  the  form  of 
a  space-lattice,  about  their  positions  of  equilibrium.  If  we 
apply  classical  statistics  to  these  vibrations,  and  particularly 
the  law  of  equipartition  of  kinetic  energy,  we  arrive  at  the 
following  conclusion  :  The  mean  kinetic  energy  of  an  atom 

01L/TT 

vibrating  in  space,  i.e.  with  three  degrees  of  freedom,  is ,  and 

its  mean  potential  energy  is  equal  to  the  same  amount,94  so 
that  its  total  energy  is  therefore  3kT.  If  we  now  consider 
1  gramme-atom  of  the  body,  that  is,  a  system  of  N  atoms 
(where  N  is  the  Avogadro  number,  approximately  6  x  1023), 
we  get  for  the  mean  energy  of  the  body,  remembering  (19), 

E  =  3kTN  =  3RT      .         .        .     (31) 
29 


30  THE  QUANTUM  THEORY 

where  R  is  the  absolute  gas-constant.  It  follows  that  the 
atomic  heat  of  the  body  at  constant  volume  becomes  : 

0,.^-3*-fr94[°£]     .        .     (32) 

This  is  the  law  of  Dulong  and  Petit,9*  according  to  which 
the  atomic  heat  (at  constant  volume)  of  monatomic  solid  bodies 

has  the  value  5'94  jj— ,  independently  of  the  temperature.96 

This  law  is  actually  obeyed  by  many  elements  more  or  less 
closely.97  On  the  other  hand,  elements  have  long  been  known 
which  are  far  from  following  this  rule,  and  which  show 
systematic  differences,  especially  at  low  temperatures. 

Thus,  as  early  as  the  year  1875,  F.  H.  Weber  &  found  that 

the  atomic  heat  of  diamond  at  -  50°  C.  is  about  0'75  ?^.     The 

deg. 

atomic  heats  of  other  elements  as  well  (boron,  beryllium, 
silicon)  have  also  been  shown  to  be  much  too  small  at 
ordinary  temperatures.  And  altogether  it  appeared  that  the 
defect  from  Dulong  and  Petit' 's  normal  value  occurs  quite 
generally  at  low  temperatures,  and  becomes  the  more  pro- 
nounced, the  lower  the  temperature.  The  classical  theory 
offered  no  solution  of  these  low  values  of  the  atomic  heat.99 

§  2.  Einstein's  Theory  of  Atomic  Heats 

Einstein  was  the  first  to  recognise  10°  that  in  this  case,  too, 
the  quantum  theory  was  destined  to  solve  the  difficulty. 
Precisely  as  in  the  theory  of  radiation,  the  method  of 
classical  statistics  leads  of  necessity  to  a  wrong  law  in  the 
field  of  atomic  heats.  Hence,  here  also,  we  must  abandon 
the  laiv  of  the  equipartilion  of  energy.  In  fact,  we  need  only 
imagine  electric  charges  distributed  among  the  atoms 101  and 
then  we  see  that,  exactly  like  the  Planck  oscillators,  they  must 
set  themselves  in  equilibrium  with  the  heat-radiation  which 
is  always  present  in  the  body.  This  means,  however,  that 

the  relation  (7),  according  to  which   U  =  ~a  K,.,  must  be  set 

up  between  the  mean  energy  U  of  an  atom  vibrating  linearly 
with  frequency  v,  and  the  intensity  of  radiation  K,,.  If  we 
now  take  Planck's  radiation  formula  (12)  as  empirically 


EINSTEIN'S  THEORY  OF  ATOMIC  HEATS     31 

given,  it  follows  immediately  that  the  mean  energy  U  of  the 
linearly  vibrating  atom  must  possess,  not  the  value  kT  given 
by  classical  statistics,  but  the  value  given  by  the  quantum 

theory,  namely,  U  = 


hv 


For  the  atom  which  vibrates 


in  space  we  get,  therefore — by  an  obvious  generalisation — in 
place  of  the  classical  value  3kT,  the  quantum  value  : 


^j 

^- 

p-- 

.. 

X 

/ 

/ 

I 

/ 

1 

/ 

. 

o    4 

1    0 

j  0.3  a, 

*  <?*  fl 

s  0,7  0,8  0,9  io                         1,5   „  i.tf    zja 
x  THr 

FIG.  3. 

The  heat-content  of  the  gramme-atom  will  therefore  be 
3Nhv 


.         .         .         .     (33) 


«w  -  1 


from  which  we  get  for  the  atomic  heat  at  constant  volume 
Einstein's  formula 


According  to  this,  the  atomic  heat  of  monatomic  solid  bodies 
is  not  a  constant  which  is  independent  of  the  temperature,  as 

Dulong  and  Petit' a  Law  requires,  but  is  a  function  of  JL  and 


32  THE  QUANTUM  THEORY 

is  therefore  in  the  case  of  a  definite  body  (i.e.  with  v  fixed) 
a  funct'ion  of  the  temperature.  Its  form  is  such  (see  Fig.  3), 
that  for  T  —  0  (i.e.  x  —  co )  the  atomic  heat  itself  is 
zero,  and  then  increases  gradually  with  increasing  tem- 
perature, approaching  asymptotically  at  high  temperatures 
(i.e.  with  small  x)  the  classical  value  3R.  Dulony  and 
Petit's  Law  is  therefore  only  true  in  the  limit  for  small 

values  of-=-jL  that  is,  for  low  frequencies  of  atomic  vibration, 

fCJ. 

or  high  temperatures,  exactly  as  is  the  case  with  Rayleigh's 
Law  of  Radiation.  The  departures  from  Dulong  and  Petit's 
Law,  in  passing  from  high  to  low  temperatures,  become  marked 
the  sooner  the  greater  the  frequency  of  the  atoms. 

§3.  Methods  of  Determining  the  Frequency 

This  frequency  v — the  only  unknown  magnitude  in  Einstein's 
formula  (34) — may  be  determined  by  several  independent  and 
very  noteworthy  methods.  One  way  that  is  always  possible 
is  of  course  the  following :  For  a  given  substance  we  choose 
an  experimentally  well-known  value  of  the  atomic  heat  C* 
which  corresponds  to  a  definite  temperature  T*.  From  (34) 

x2ex          C* 

it  follows  then  that —  =  ~,  an  equation  from  which 

(e*  —  1)        oH 

x  =  — ^  can  be  determined,  and  thence  v.     From  the  v  thus 

found  the  course  of  the  whole  Gv  curve  can  be  calculated  for 
all  temperatures,  and  compared  with  experiment. 

Besides  this  "empirical"  method  of  determining  v,  there 
are  a  number  of  other  more  "  theoretical "  methods  which 
do  not  require  the  use  of  the  values  of  the  atomic  heat. 
Einstein,102  as  far  back  as  1911,  discovered  an  important 
connection  between  the  frequency  v  and  the  elastic  properties 
of  the  body.  That  such  a  connexion  must  exist  is  easily 
recognised  from  the  following  considerations :  imagine  the 
atoms  of  the  body  arranged  upon  a  space-lattice,  as  in  a 
crystal,  and  suppose  a  certain  definite  atom  arbitrarily  dis- 
turbed from  its  position  of  rest,  then  this  atom,  when  released, 
will  execute  vibrations  about  its  position  of  equilibrium.  If 
we  suppose  these  vibrations  to  be  simply  periodic  ("  mono- 
chromatic")— we  shall,  however,  soon  recognise  that  this 


DETERMINING  THE  FREQUENCY    33 

supposition  is  an  inadmissible  approximation  —  we  see  that 
the  frequency  v  is  the  greater  the  smaller  the  atomic  mass, 
and  therefore  also  the  atomic  weight  of  the  body,  and  the 
greater  on  the  other  hand  the  force  which  restores  the 
atom  to  its  position  of  equilibrium.  This  restoring  force  is, 
however,  for  its  part  the  stronger,  the  less  extensible  and 
therefore  compressible  the  body  is.  Hence  v  must  turn  out 
the  greater,  the  smaller  the  atomic  weight  and  the  compres- 
sibility of  the  substance.  The  exact  working  out  of  this  idea 
led  Einstein  to  the  formula  103 

2-8.  107  ,„.. 

....    (35) 


Where  A  is  the  atomic  weight,  p  the  density,  and  *  the 
compressibility  of  the  body. 

A  further  interesting  relation,  which  connects  v  with  ther- 
mal data,  namely,  the  melting-point,  was  found  by  F.  A. 
Lindemann  1M  by  working  out  the  conception  that  the  ampli- 
tude of  vibration  of  the  atom  at  the  melting-point  is  of  the 
order  of  magnitude  of  the  distances  between  the  atoms.  If 
Ts  is  the  absolute  melting-point,  then  it  follows  that 

.         .         .     (36) 

Another  formula  deduced  by  E.  Gruneisen  «»  may  also  be 
given  here  : 

v  =  2-91.10".^-^[cj.a-i.pi]0    .         .     (37) 

Here  Cv  is  the  atomic  heat  at  constant  volume,  and  a  is  the 
coefficient  of  thermal  expansion  ;  the  index  0  means  that  the 
value  of  C\a.-*pk  at  absolute  zero  is  to  be  used. 

From  formulae  (35)  and  (36)  we  recognise  at  once  the 
abnormal  behaviour  of  diamond,  for  example,  in  respect  to  its 
atomic  heat.  For  it  is  known  that  diamond  has  a  high  melt- 
ing-point and  very  low  compressibility  accompanied  by  a  low 
atomic  weight.  Its  v  is  therefore  comparatively  large,  and 
it  follows  therefore,  according  to  the  above  considerations, 
that  its  atomic  heat  falls  below  Dulong  and  Petit'  s  value  of 

3B  =  5-94    —  at  comparatively  high  temperatures.    In  fact 
3 


34  THE  QUANTUM  THEORY 

the  atomic  heat  of  diamond  at  284°  abs.  is  only  T35  -? — '-,  at 

413°  abs.  it  is  3'64  ~~ '  and  even  at  1169°  abs.  it  reaches  only 
deg.' 

the  value  5'24  —• 
deg. 

Finally,  particular  importance  attaches  to  a  relation,  first 
discovered  by  E.  Madelung 106  and  W.  Sutherland,™  between 
the  frequency  v  of  the  atoms  and  the  optical  properties 
of  bodies.  The  two  investigators  started  in  this  case  from 
the  following  conception  :  Crystals  of  diatomic  compounds 
(binary  salts),  such  as  rock-salt  (NaCl),  sylvin  (KC1), 
potassium  bromide  (KBr),  and  others,  are  known  to  be 
cubical  space-lattices,  in  which  the  single  atoms  carry  electric 
charges,  and  therefore  appear  as  ions.  In  fact,  the  points  of 
the  space-lattice  are  occupied  alternately  by  the  positively 
charged  Na+  (or  K+)  atoms,  and  the  negatively  charged  Cl~ 
(or  Br~)  atoms.  If  an  electromagnetic  light- wave  of  frequency 
v  falls  upon  this  crystal,  the  two  ions  are  thrown  into  forced 
oscillations  relatively  to  one  another,  and  further,  on  account 
of  "  resonance,"  the  more  strongly,  the  more  exactly  the  fre- 
quency v  of  the  impinging  wave  agrees  with  the  natural  fre- 
quency vr,  which  lies  in  the  infra-red,  of  the  ions  themselves. 
Since  the  ionic  vibrations  are  set  up  at  the  cost  of  the  energy 
of  the  impinging  wave,  this  energy  will  be  weakened  (ab- 
sorbed) the  more  during  its  passage  through  the  body,  the 
nearer  v  lies  to  vr.  On  the  other  hand,  the  vibrating  ions 
radiate  back  waves  of  frequency  v  since  they  are  compelled 
to  execute  these  vibrations,  when  set  into  forced  vibration, 
doing  so  the  more  strongly,  the  more  pronounced  the  reson- 
ance is,  again,  therefore,  the  nearer  v  lies  to  vr.  Hence  a 
region  of  maximum  absorption  and  strongest  (metallic)  re- 
flection will  lie  in  the  neighbourhood 108  of  v  =  vr.  These 
regions  of  metallic  reflection  of  a  given  substance  may  be 
detected  by  the  method  of  "  Eeststrahlen "  (residual  rays) 
worked  out  by  H.  Rubens  and  E.  F.  Nichols.'109  For  this 
purpose  we  only  require  to  reflect  radiation  of  a  considerable 
range  of  frequency  about  v  repeatedly  from  the  substance. 
In  this  way  all  waves  will  be  gradually  absorbed  except  those 
most  strongly  reflected.  These  are,  however,  just  those  of 


NERNST'S  HEAT  THEOREM  35 

frequency  vr.  They  are  thus  "residual."  The  ultra-red  fre- 
quency vf  of  the  ions  therefore  agrees  with  the  frequency  of 
the  residual  rays.110  On  the  other  hand,  this  vibration  of  the 
charged  atoms  is  dependent  on  the  elastic  properties  of  the 
substance,  as  we  recognised  in  considering  the  formula  (35). 
We  thus  conclude  that  the  "  elastic"  frequency  of  the  atoms 
of  binary  salts  agrees  to  a  close  approximation  with  the 
"optical"  frequency  of  their  residual  rays.  But  since  the 
"  elastic  "  frequency  of  the  atoms  determines  the  behaviour 
of  their  atomic  heat,  the  ring  is  thereby  closed,  and  W. 
Nernst m  was  thus  justified  in  propounding  the  fundamental 
law,  that  in  calculating  the  atomic  heat  of  binary  salts,  we 
may  simply  insert  for  the  atomic  frequencies  v  the  frequencies 
of  the  residual  rays. 

In  this  way  a  number  of  independent  ways  were  opened 
up  for  determining  the  atomic  frequencies  required  for  the 
calculation  of  the  atomic  heat.  A  comparison  of  the  various 
values  of  v  determined  by  these  different  methods  shows  in 
general  satisfactory  agreement,  at  any  rate  in  order  of  magni- 
tude.112 One  could  hardly  expect  more,  as  we  shall  soon  see, 
in  view  of  the  many  idealised  conditions  that  were  used  in 
the  theory. 

§4.  Nernst's  Heat  Theorem 

With  a  view  to  discovering  experimentally  the  general  law 
for  the  decrease  of  the  atomic  heat  when  approaching  low  tem- 
peratures W.  Nernst 113  began  in  1910,  in  co-operation  with 
his  research  students,  a  series  of  masterly  and  widely  planned 
researches.  For,  by  an  entirely  different  route  from  Einstein 
— namely,  by  way  of  thermodynamics — he  also  had  become 
convinced  that  the  atomic  heat  of  solid  bodies  must  become 
vanishingly  small  on  approaching  absolute  zero.  In  his 
opinion  this  result  was  only  one  of  several  consequences  of  a 
general  principle,  namely,  a  new  law  of  heat.11*  This  Heat 
Theorem  of  Nernst — often  called  the  Third  Law  of  Thermo- 
dynamics— states,  in  its  original  form,  the  following  fact :  If 
we  regard  a  system  of  condensed  (i.e.  liquid  or  solid)  bodies, 
which  passes  at  temperature  T  by  means  of  an  isothermal 
reaction  from  one  state  to  another,  and  if  A  is  the  maximum 
work  which  can  be  gained  from  this  reaction,  then 


36  THE  QUANTUM  THEORY 

gy-0  for  the  limit  T  =  0          .         .     (38) 

tliat  is  to  say,  in  the  immediate  neighbourhood  of  absolute  zero, 
the  maximum  work  which  can  be  gained  is  independent  of  the 
temperature  But  it  follows  immediately  from  this,  if  we 
apply  the  two  laws  of  thermodynamics,119  that  for  any 
reaction  which  changes  the  system  from  the  initial  condition 
with  energy  U^  to  the  final  condition  with  energy  U.2,  the 
relation  holds  that 

^-^for  the  limit  !T»0      .         .     (39) 

Now,  since  -5^,  if  we  take  a  gramme-atom  of  the  substance, 

gives  the  atomic  heat,  we  are  led  to  enunciate  the  following 
rule :  in  the  immediate  neighbourhood  of  absolute  zero,  the 
atomic  heat  of  condensed  systems  remains  unchanged  during 
any  transformation. 

Planck  m  has  given  Nernst's  Theorem  a  still  more  general 
form :  Not  only  the  difference  of  the  atomic  heats  (before  and 
after  the  reaction)  is  to  assume  the  value  0  at  absolute  zero,  but 
also  each  atomic  heat  itself  is  to  do  the  same.  Thus  it  follows 
from  the  extended  Nernst  Theorem,  in  agreement  with  the 
demands  of  the  quantum  theory,  that  the  atomic  heats  of 
solid  bodies  disappear  at  absolute  zero. 

§  5.  The  Improvement  on  Einstein's  Theory  of  Atomic  Heats 

The  experiments  of  Nernst  and  his  collaborators  proved 
quite  convincingly  that  the  atomic  heat  of  all  solid  bodies 
tends  towards  a  zero  value  as  the  temperature  falls.  In 
the  main,  the  courses  of  these  decreasing  values  showed  a 
notable  agreement  with  Einstein's  formula  (34).  At  low 
temperatures,  however,  systematic  discrepancies  were  found 
in  all  cases,  in  the  sense  that  the  observed  atomic  heats  fell 
off  much  more  slowly  than  Einstein's  formula  demanded.1" 
W.  Nernst  and  F.  A.  Lindemann  "8  tried  to  take  these  dis- 
crepancies into  account  by  constructing  an  empirical  formula, 
and  this  actually  expressed  the  observations  much  more 
accurately  than  did  the  Einstein  formula.  This  Nernst- 


IMPROVEMENT  ON  EINSTEIN'S  THEORY     37 

Lindemann  formula,  which  is  now  only  of  historical  interest, 
is  as  follows  : — 


hv 
where  x  —  .     (40) 


It  receives  a  meaning  if  we  suppose  that  one  half  of  all  the 
atoms  vibrate  with  the  frequency  v,  the  other  half  with  the 

frequency    n  •    While  this  supposition  is  untenable  in  this 

raw  form,  it  contains  a  kernel  of  truth,  namely,  recognition 
of  the  fact  that  the  "  monochromatic "  theory  of  atomic 
heats,  which  assumes  only  a  single  fixed  frequency  v  for  all 
atoms,  goes  too  far,  being  an  idealisation  of  the  real  state  of 
affairs.  Einstein,  who  at  first,  for  the  sake  of  simplicity, 
reckoned  with  only  one  frequency,  had  himself  already 
recognised  how  matters  stood,  and  drawn  attention  to  the 
need  for  amending  his  theory.119  Nowadays,  in  fact,  we  think 
of  a  solid  body,  say  a  crystal,  as  built  up  of  atoms  regularly 
arranged  upon  a  space-lattice,  according  to  Bravais'  concep- 
tion ;  and  this  hypothesis  has  been  verified  as  a  certainty 
through  Lane's  discovery  of  the  interference  of  Rontgen  rays. 
In  such  a  complicated  mechanical  system,  however,  the 
single  atoms  do  not  vibrate  independently  of  one  another 
with  a  single,  frequency  v.  But  the  position  of  equilibrium 
of  each  atom,  and  thereby  the  type  of  its  oscillations  about 
that  position,  is  determined  rather  by  the  forces  which  all  the 
other  atoms  of  the  body  exert  upon  the  atom  in  question. 
We  are  confronted  with  a  structure  which  is  comparable  to 
the  one-dimensional  case  of  a  vibrating  string,  and  which 
thus  possesses  a  whole  spectrum  of  natural  frequencies, 
corresponding  to  the  overtones  of  the  string.  If  the  body 
consists  of  N  atoms,  it  possesses  in  general  3N  natural 
frequencies,120  of  which  the  slowest  are  sound  waves,  while 
the  quickest  fall  in  the  infra-red.  The  most  general  possible 
movement  of  each  atom  then  consists  in  a  super-position 
of  all  these  natural  frequencies.  Now,  since  each  natural 
frequency  represents  a  linear,  i.e.  simple  periodic,  motion, 
exactly  like  the  motion  of  a  Planck  oscillator,  the  idea 


66038 


88  THE  QUANTUM  THEORY 

naturally  suggested  itself,  in  calculating  the  energy-content 
of  the  body,  to  allot  to  each  natural  frequency  of  period  v  the 

hv 
theoretical  quantum  amount  -^ as  if  the  natural  period 

eFf-l 

were   identical   with  a  linear  oscillator.      The   total   mean 
energy  of  the  body  then  becomes 


in  which  the  summation  is  carried  over  all  3N  natural 
frequencies  vv  v2,  v3,  .  .  .  v3jv,  that  is,  over  the  whole  elastic 
spectrum  of  the  substance.  By  differentiation  with  respect 
to  T  we  obtain  the  atomic  heat 


§  6.  Debye's  Theory  of  Atomic  Heats 

The  kernel  of  the  problem  thus  consists  in  calculating  the 
"  elastic  spectrum  "  of  a  given  body,  that  is,  in  determining 
for  any  body  the  position  of  its  natural  periods.  In  this  sense, 
the  theory  has  been  worked  out  from  two  different  sides ;  on 
the  one  hand  by  P.  Debye,nl  who  took  an  elastic  continuum 
as  an  approximation  to  the  actual  atomically  constructed 
body,  and  on  the  other  by  M.  Born  and  v.  Kdrmdn,™  who 
replaced  the  crystal  of  limited  size  by  one  of  infinite  di- 
mensions. The  difference  between  these  two  methods  of 
approximation  causes  the  main  problem,  namely,  the  working- 
out  of  the  elastic  spectrum,  to  be  solved  quite  differently  in 
the  two  cases.  The  Debye  theory,  which  from  the  outset 
leaves  out  of  consideration  the  crystalline,  and  even  the 
atomic,  structure  of  the  body,  rests  upon  the  classical  theory 
of  elasticity,  which,  of  course,  treats  bodies  as  structureless 
continua.  From  it  follows  the  important  law :  the  number 
Z(v)dv  of  all  those  natural  periods,  the  frequency  of  which 
falls  within  the  interval  v,  v  +  dv,  amounts  to 123 


DEBYE'S  THEORY  OF  ATOMIC  HEATS      39 


Here  V  is  the  volume  of  the  body,  GI  and  ct  are  the  velocities 
with  which  longitudinal  and  transverse  waves,  respectively, 
are  propagated  within  the  body.  In  this  case,  however,  the 
following  difficulty  occurs  in  replacing  the  body,  which  in 
reality  consists  of  N  atoms,  by  a  continuum,  namely,  the 
elastic  spectrum  extends  to  infinity,  that  is,  the  number  of 
natural  frequencies  becomes  infinitely  great.  For  example, 
the  number  of  natural  frequencies  (fundamental  tone  and 
over-tones)  of  a  linear  string  of  length  L  are 

Vi  =  ct  •  ^f  and  vi  =  GI  •  ^j.  respectively   (i  =  1,  2,  .  .  .  x>  ) 

2-Lv  a±J 

according  as  to  whether  we  are  considering  transverse  or 
longitudinal  frequencies.  The  series  of  overtones  therefore 
extends  without  limit  to  infinity.  In  reality,  however,  as  the 
body  consists  of  N  atoms  (mass-points),  it  may  not  possess 
more  than  3N  natural  frequencies.  In  order  to  attain  this, 
Debye  helps  himself  out  by  means  of  the  following  bold 
supposition.  Instead  of  calculating  strictly  the  elastic  spec- 
trum of  the  real  body  consisting  of  N  atoms,  he  replaces  it  by 
that  of  the  continuum  as  an  approximation,  but  breaks  it  off 
arbitrarily  at  the  3Nth  natural  period.  Debye  thus  gets  the 
greatest  frequency  vm  which  occurs,  that  is,  the  upper  limit 
of  the  elastic  spectrum,  from  the  condition  : 


0 


therefore 


[9JV         I 
"(H) 


.     (44) 


The  atomic  heat  of  the  body,  which  follows  from  (42),  is 


40  THE  QUANTUM  THEORY 

a  result  which  can  easily  be  brought  into  the  following  more 
simple  form  :  1M 


The  atomic  heat  is  therefore  only  a  function  of  the  magnitude 
xm,  that  is,  it  depends  only  on  the  ratio  ^  :  here  ^  =  ^ 

This  result  may  be  expressed  in  Debye's  terms  thus  :  reckon- 
ing the  temperature  T  as  a  multiple  of  a  temperature  ®  which 
is  characteristic  of  the  particular  body,  then  the  atomic  heat  is 
represented  for  all  monatomic  bodies  by  the  same  curve.  Hence 
we  must  be  able  to  bring  the  C0  curves  of  all  monatomic 
bodies  into  coincidence,  if  only  the  scale  of  temperature  be 
suitably  chosen  for  each  substance.128  For  high  tempera- 
tures, the  Debye  formula  passes  over,  as  it  must  do,  into  the 
classical  value  of  Dulong  and  Petit,  Cv  =  3.R,128  just  as  do 
the  Einstein  and  Nernst-Lindemann  formulae.  On  the  other 
hand,  it  differs  from  these  latter  in  falling  much  more  slowly 
at  low  temperatures.  For  while  the  atomic  heats,  according 
to  both  Einstein  and  Nernst-Lindemann,  fall  exponentially 

/  1  constN 

(with  7™-e~~T~)  at  low  temperatures,  Debye's  formula  leads 

to  the  fundamental  law,127  that  the  atomic  heats  of  all  bodies  at 
low  temperatures  are  proportional  to  the  third  power  of  the 
absolute  temperature. 

It  is  further  remarkable,  that  we  may  write  formula  (44) 
for  the  maximum  natural  frequency  in  a  form  such  that  only 
measurable  magnitudes  occur  in  it.  For  if  we  express  the  two 
velocities  of  sound  Ct  and  c\  in  terms  of  the  elastic  constants  of 
the  body,  and  replace  the  volume  Fof  the  gramme-atom  by 

,.     .  atomic  weight   (A)   . 
the  quotient  --  densityg(p)  (   >,  *  follows  that« 

,5-28.10*.^) 
where  "         ' 


DEBYE'S  THEORY  OF  ATOMIC  HEATS      41 

In  it  K  is  again  the  compressibility  of  the  body,  <r  the 
Poisson  ratio,  that  is,  the  ratio  of  the  transverse  contraction 
to  the  extension.  The  similarity  of  this  formula  with  the 
Einstein  relation  (35)  strikes  one  immediately.  But  in  this 
case  the  second  elastic  constant  of  the  isotropic  body,  cr, 
enters  into  the  equation  as  well.  Altogether,  the  upper  limit 
vm  of  the  elastic  spectrum,  at  which,  as  one  can  show,129  the 
natural  frequencies  always  crowd  together  closely,  plays  in 
the  stricter  theory  an  analogous  role  to  that  played  by  the 
single  natural  frequency  v  in  the  "  monochromatic  "  theory. 

Comparison  with  experiment  shows130  that  the  Debye 
formula,  at  any  rate  for  the  monatomic  elements  such  as 
aluminium,  copper,  silver,  lead,  mercury,  zinc,  diamond,  de- 
scribes the  course  of  values  of  the  measured  atomic  heats 
very  accurately.  Particularly  at  low  temperatures,  the  pro- 
portionality between  the  atomic  heat  and  the  third  power  of 
the  absolute  temperature  receives  fair  confirmation.131  In 
view  of  the  fact  that  the  idealised  view  (replacement  of  the 
actually  atomic  body  by  a  continuum)  is  carried  very  far,  we 
must  not  regard  the  agreement  between  theory  and  experi- 
ment as  self-evident.  At  low  temperatures,  Debye's  idealis- 
ation will  justify  itself.  For  then  ^  is  large,  and  hence  the 

amount  of  energy         v —  is  small,  excepting  when  v  itself  as- 

ekf  -  1 

sumes  small  values.  At  low  temperatures,  there/ore,  only  long 
waves  will  contribute  sensibly  to  the  energy  of  a  body,  and  hence 
to  its  atomic  heat.  For  long  waves,  however,  that  is,  for 
waves,  the  length  of  which  is  great  compared  with  the  dis- 
tance between  the  atoms,  the  specific  atomistic  construction  of 
the  body  plays  no  part ;  for  them  the  substance  is  almost  a 
continuum.  The  position  is  quite  different  at  high  tempera- 
tures, at  which  the  longer  frequencies  up  to  the  maximum  vm 
(that  is,  the  shorter  waves  down  to  the  smallest)  furnish  con- 
tributions of  energy.  For  the  waves  which  correspond  to  the 
highest  frequencies  possess  lengths,  as  can  easily  be  shown,132 
which  are  comparable  with  the  distances  between  the  atoms, 
and  for  these  shorter  waves  the  medium  cannot  fail  to  betray 
its  atomic  structure.  Here,  therefore,  its  replacement  by  a 


42  THE  QUANTUM  THEORY 

continuum  becomes  questionable  since  the  approximation  is 
only  very  rough. 

§  7.  The  Lattice  Theory  of  Atomic  Heats  according  to  Born  and 
Karman.    The  Elastic  Spectrum  of  the  most  general  Crystal 

At  this  point  the  above-mentioned  investigations  of  Born 
and  Kdrmdn  intervene,  which,  going  beyond  Debye,  take 
account  of  the  real  crystalline  structure  of  the  body,  that  is 
to  say,  the  space-lattice  arrangement  of  the  atoms.  In  order 
to  overcome  the  great  mathematical  difficulties  involved,  they 
imagined,  as  has  already  been  said,  the  actual  limited  crystal 
replaced  by  one  extended  indefinitely.  Thus  the  disturbing 
effect  of  the  surface  on  the  interior  could  be  eliminated,  so 
that  now  all  atoms  were  exposed  to  the  same  conditions. 
Here  also  the  main  problem  is  again  to  determine  the  elastic 
spectrum,  or — if  we  dispense  with  the  exact  calculation  of 
the  proper  frequencies — at  least  to  discover  the  law,  accord- 
ing to  which  the  proper  (or  natural)  frequencies  are  distributed 
among  the  different  regions  of  frequency.  This  problem  was 
first  solved  by  Born  and  Kdrmdn  for  regular  crystals.  The 
laws  thus  obtained  were  then  extended  to  the  case  of  simple 
point-lattices  of  arbitrary  symmetry,  and  finally,  Born  de- 
duced them,  in  his  "Dynamics  of  the  Crystal  Lattice,"  for 
the  most  general  form  of  space-lattice.133 

These  most  general  space-lattices  arise  from  the  periodic 
repetition  in  space  of  a  definite  group  of  atoms  and  electrons 
(basic  group)  which  on  the  whole  is  electrically  neutral,  and 
is  enclosed  in  a  parallelepiped  of  space,  the  "elementary 
parallelepiped."  In  Fig.  4  such  a  lattice,  in  this  case,  how- 
ever, plane,  is  illustrated,  in  which  the  basic  group  consists 
of  three  particles  (-ox).  All  •  particles  form  together  a 
simple  lattice,  as  do  the  o  and  x  particles.  We  have  in  this 
way  three  interlocked  simple  lattices. 

Thus,  for  example,  the  halogen  compounds  of  the  alkalies 
(NaCl,  LiCl,  KC1,  KBr,  KI,  BbCl,  EbBr,  Ebl,  and  so  forth) 
form  cubic  space-lattices,  in  which  the  lattice  points  are 
alternately  occupied  by  the  positive  alkali  ion  and  the  negative 
halogen  ion  (see  Fig.  5).  If  we  regard  the  whole  cube  here 
pictured  as  the  "  elementary  cube,"  then  the  basic  group  would 


LATTICE  THEORY  OF  ATOMIC  HEATS     43 

contain  eight  particles,  namely,  four  ions  of  each  sort  (they  are 
numbered  here).  We  have  thus  eight  interpenetrating  simple 
lattices.  Every  four  of  them  would,  however,  consist  of  the 
same  kind  of  particle.  Hence  it  is  advisable  to  select  in  this 
case  in  place  of  the  cube  the  rhombohedron  (double-lined  in 


FIG.  4. 

the  figure)  as  the  elementary  parallelepiped.  Then  the  basic 
group  consists  only  of  the  two  different  particles  1  and  8,  of 
which  the  one  lies  in  a  corner,  the  other  in  the  middle  of  the 
parallelepiped.  In  fact  we  can  get  the  whole  lattice  by  displac- 
ing the  basic  group  in  the  direction  of  the  three  rhombohedral 
edges,  a  distance  equal  to  a  whole  multiple  of  the  length  of 


44  THE  QUANTUM  THEORY 

the  edge.  The  lattice  consists  therefore,  according  to  this 
view,  of  two  interlaced  simple  cubical  atomic  lattices.  Further- 
more, they  are  "  surface-centred  "  lattices,  that  is  to  say,  such 
that  not  only  the  corners  of  the  cubes,  but  also  the  middle 
points  of  the  cube-surfaces,  are  occupied.  If  in  the  most 
general  case  the  basic  group  contains  s  different  particles,  the 
lattice  consists  of  s  interlaced  simple  lattices. 


FIG.  5. 

In  order  now  to  get  a  general  view  of  the  laws  which 
govern  the  elastic  spectrum  of  such  a  most  general  crystal, 
we  proceed  according  to  Born  and  Kdrmdn  as  follows :  We 
imagine  an  elastic  wave  of  definite  wave-length  and  definite 
direction  (the  normal  to  the  wave  front)  passing  through  the 
crystal.  For  each  wave  thus  defined  there  are  3s  natural 


LATTICE  THEORY  OF  ATOMIC  HEATS     45 

frequencies  with  periodicities  vl  v2  v3  .  .  .  v3.«.  The  first  three 
frequencies  vlt  v.,,  v3  correspond  to  those  natural  frequencies 
of  the  crystal,  by  which  the  single  interpenetrating  simple 
lattices  are  similarly  distorted  to  a  first  approximation  with- 
out being  compelled  to  move  relatively  to  one  another. 
These  are  the  three  ordinary  acoustic  natural  periods  (one 
longitudinal,  two  transverse).  The  remaining  3(s  -  1) 
frequencies,  on  the  other  hand,  correspond  to  another  type 
of  motion  of  the  crystal,  namely,  to  those  natural  frequencies 
with  which  the  single  simple  lattices  oscillate  with  respect  to 
one  another  without  distortion.  If  the  basic  group  contains 
only  one  particle  (s  =  1),  i.e.  if  the  crystal  consists  of  only  a 
simple  lattice,  this  second  type  of  motion  disappears  alto- 
gether, and  we  are  left  with  only  the  three  acoustic  natural 
frequencies  vv  v.2,  v3.  If,  on  the  other  hand,  we  are  dealing 
with  a  crystal,  say  of  the  halogen  compounds  of  an  alkali, 
for  example,  rock-salt  (NaCl),  s  =  2,  there  exist,  as  we  have 
seen,  besides  the  three  acoustic  oscillations,  three  further 
natural  frequencies  of  the  second  type.  In  consequence 
of  the  regular  crystal  character  of  the  alkaline  halides,  these 
three  natural  frequencies  exactly  coincide,  at  any  rate  for 
long  waves,  and  give  rise  to  that  motion  in  which  the  sodium 
lattice  vibrates  approximately  as  a  rigid  structure  against  the 
likewise  rigid  chlorine  lattice.  We  see  at  once  that  it  is 
just  the  natural  frequency  last  considered  that  will  play  the 
chief  part  in  the  optics  of  these  crystals.  For  when  an 
electromagnetic  wave  meets  the  crystal,  the  sodium  ions 
are  driven  by  the  electric  force  of  the  wave  to  the  one  side, 
and  the  oppositely-charged  chlorine  atoms  are  drawn  to  the 
opposite  side.  It  is  thus  just  the  type  of  vibration  described 
above  that  is  brought  about.  If  the  frequency  of  the 
external  wave  approaches  closely  to  that  of  the  natural 
period,  resonance  occurs.  These  infra-red  vibrations,  there- 
fore, are  what  determine  the  course  of  the  refractive  index, 
especially  in  the  infra-red.  They  are  the  so-called  "  infra- 
red dispersion  frequencies."  It  is  also  in  their  neighbourhood 
that  the  places  of  metallic  reflection  lie  which  are  detected 
by  the  method  of  residual-rays. 

What   has   just   been   stated   for   the   special   case  s  =  2 
(alkaline  halides)  may,  of  course,  be  immediately  generalised. 


46  THE  QUANTUM  THEORY 

For  if  the  basic  groups  consists  of  s  different  particles,  it 
is  just  the  3(s  -  1)  natural  frequencies  that  determine  the 
dispersion  of  the  crystal.  Among  them  are  those,  in  the 
neighbourhood  of  which  the  regions  of  metallic  reflection 
(residual  rays)  lie.  If  the  basic  group  contains  p  positive 
atomic  residues  and  s  -  p  electrons,  the  frequencies  v4  .  .  .  v3, 
fall  correspondingly  into  two  classes  :  the  first  class  consists 
of  3(p  -  1)  infra-red  frequencies,  which  arise  from  the 
atomic  residues ;  the  second  consists  of  3(s  -  p)  ultra-violet 
frequencies,  which  are  to  be  ascribed  to  the  influence  of  the 
electrons.  The  infra-red  natural  frequencies  decide  the 
course  of  the  refractive  index  in  the  infra-red,  the  position 
of  the  residual  rays,  and,  as  we  shall  see,  the  atomic  heats ; 
the  ultra-violet  natural  frequencies,  on  the  other  hand,  deter- 
mine chiefly  the  refractive  indices  in  the  visible  and  ultra- 
violet. Incidentally,  the  general  lattice-theory  of  Born 134 
confirms  the  law  previously  enunciated  by  Haber 13S  that  the 
frequencies  of  the  first  class  (infra-red)  bear  the  same  ratio  to 
the  second  (ultra-violet)  class,  as  regards  order  of  magnitude, 
as  the  square  root  of  the  mass  of  the  electron  bears  to  the 
square  root  of  the  mass  of  the  atom. 

After  this  digression  let  us  now  return  to  our  starting-point. 
Up  to  the  present  we  have  always  considered  a  wave  of 
definite  length  A.  and  with  a  definite  normal  direction  n,  and 
we  have  seen  that  corresponding  to  it  there  are,  in  the  most 
general  case,  3s  natural  frequencies  i/j  .  .  .  v3S.  Let  us  now 
allow  the  wave-length  X  to  vary  continuously,  keeping  the 
wave-direction  constant,  by  going  from  infinitely  long  waves 
to  the  smallest.  Then  each  of  the  3s  natural  frequencies  will 
also  vary  continuously,  and  will  pass  through  a  continuous 
range  of  values.  In  other  words,  the  3s  natural  frequencies 
are  certain  functions  of  the  wave-length  X  : 

v< -/<(*). 

From  this,  however,  we  learn  the  fundamental  fact  that  all 
these  ranges  of  values  of  the  single  natural  frequencies  are 
only  finite  in  extent  and  that,  therefore,  each  of  the  3s  continua 
of  frequencies  automatically  breaks  off  at  a  higJiest  limiting 
frequency.  "  Automatically,"  i.e.  without  our  arbitrary  assist- 
ance (as  in  Debye's  case),  solely  on  account  of  the  analytical 


LATTICE  THEORY  OF  ATOMIC  HEATS     47 

form  of  the  function  /,-.  This  is  explained  by  the  fact  that 
the  wave-length  X  of  possible  waves  in  the  crystal  has  a 
lower  limit  set  to  it :  waves  of  length  below  a  certain  lowest 
value  cannot  exist.  This  is  most  simply  recognised  from  the 
following  instructive  example.  If  we  consider  a  simple 
cubical  lattice  having  the  atomic  distance  a,  and  examine, 
for  example,  longitudinal  waves,  which  are  being  propagated 
along  an  edge  of  the  cube — so  that  all  atoms  on  an  edge  at 
light  angles  to  this  side  oscillate  in  the  same  phase  in  the 
direction  of  the  edge — then  we  see  at  once  that  the  smallest 
wave  that  is  possible  here  has  the  length  Xmjn  =  2a.  For 
this  wave,  namely,  successive  planes  of  the  cube  swing 
in  opposite  phase,  that  is,  "  against "  one  another.  The 
functional  relation  between  v  and  X  assumes  the  special 
form  : 138 


For  infinitely  long  waves  (X  =  co  ),  v  =  0  ;  if  we  pass  on 
to  shorter  waves,  v  increases  continuously,  until,  for  X  =  2a, 
it  reaches  its  maximum  value  vm.  At  this  limiting  frequency 
vm  the  range  of  possible  v's  breaks  off  automatically. 

Up  to  the  present  we  have  given  the  wave-direction  (n,  the 
direction  of  the  normal)  a  certain  fixed  value,  and  have 
allowed  the  wave-length  X  to  vary.  We  now  give  the  wave- 
direction  by  degrees  other  values,  and  at  each  step  we  allow 
the  wave-length  to  vary  from  the  value  GO  to  the  least 
possible  value.  Then  the  nature  of  the  functional  dependence 
of  the  magnitude  vt  or  X,  and  the  position  of  the  limiting 
frequencies  also  change  continuously  with  the  wave-direction, 
so  that  we  may  say :  the  3s  natural  frequencies  are,  in 
general,  continuous  functions  of  the  wave-length  X  and  of 
the  wave-direction  n : 

vi  =/<(X,  n),         (i  =  1,  2,  3,  .  .  .  3s)          .     (48) 

In  it,  each  of  the  functions  /t-  breaks  off  automatically  for  a 
minimum  value  of  the  wave-length  at  an  upper  limit 
(vi)max,  which  itself  still  depends  on  the  wave-direction. 
These  equations  express  the  law  of  dispersion  of  waves  in 
crystals,  for  they  determine  for  each  wave  the  3s  frequencies 


48  THE  QUANTUM  THEORY 

vf  and  hence  also  tell  us  how  the  rates  of  propagation 
qf  =  vi'X  depend  on  the  wave-length  and  the  wave-direction. 
The  dispersion  law  becomes  particularly  simple  in  the  region 
of  long  waves :  for  the  three  acoustic  vibrations  the  re- 
lations 137 

A  A.  A. 

hold.  In  them  the  three  magnitudes  q^(n),  q.2(n),  and  qs(n) 
are  three,  in  general  different,  functions  of  the  wave- 
direction.  And  further,  these  are  the  three  velocities  of 
propagation  of  the  three  acoustic  vibrations.  In  the  region 
of  long  waves,  therefore,  the  three  velocities  of  propagation 
of  the  three  slow  acoustic  vibrations  are  independent  of  the 
wave-length  to  a  first  approximation. 

The  dispersion  law  (for  long  waves)  assumes  a  very 
different  appearance  for  the  3(s  -  1)  rapid  vibrations 
v.,  VK  .  .  v,..  It  assumes  the  form 


(i  =  4,  5,  .  .  .  3s)         .     (50) 

A 

here  the  v9's  are  constants,  the  p»(w)'s  are  again  certain 
functions  of  the  wave-direction.  The  velocities  of  propaga- 
tion here  assume  the  values 

qt  =  Vi\  =  v°\  +  p{(n)     .         .         .     (51) 

and  would  thus  be  linear  functions  of  the  wave-length. 

We  may  summarise  thus :  the  elastic  spectrum  of  the  most 
general  crystal,  the  basic  group  of  which  contains  s  particles, 
consists  of  3s  separate  parts  ("  branches  ").  Each  part  consists 
of  a  finitely  extended  continuum  of  frequencies.  The  three  first 
parts  contain  the  totality  of  all  sloiv,  acoustic  natural  fre- 
quencies (sometimes  called  "characteristic").  The  remaining 
3(s  -  1)  parts  include  the  rapid  (infra-red  and  ultra-violet) 
natural  frequencies,  which  play  the  chief  part  in  determining 
the  optical  dispersion  and  the  positions  of  metallic  reflection. 

%  8.  Continuation.     The  Law  of  Distribution  of  the  Natural 
Frequencies 

While  this  knowledge  of  the  general  character  of  the  elastic 
spectrum  is,  as  we  shall  soon  see,  of  great  value,  it  is  none 


LATTICE  THEORY  OF  ATOMIC  HEATS     49 

the  less  insufficient  for  the  question  of  the  energy-content  and 
molecular  heat  of  the  crystal,  inasmuch  as,  even  for  the 
simplest  crystal,  a  strict  calculation  of  the  elastic  spectrum 
is  not  possible  at  the  present  time.  We  know,  however,  on 
the  other  hand,  that  we  need  not  know  the  whole  details  of 
the  elastic  spectrum  to  calculate  the  energy-content  and  the 
molecular  heat,  but  that  it  suffices  to  know  the  law  according 
to  which  the  natural  frequencies  are  distributed  over  the 
elastic  spectrum  (or  its  individual  "branches").  This  is 
the  more  true,  the  closer  together  the  natural  frequencies 
lie.  Now,  in  reality  the  finite  crystal  possesses,  if  it  con- 
sists of  the  basic  group  (of  s  particles)  N  times  repeated, 
SNs  natural  frequencies,  which  are  distributed  so  that  N  fre- 
quencies fall  to  each  of  the  3s  branches  of  the  spectrum.  If  N 
becomes  infinite,  the  N  individual  natural  frequencies  of  each 
branch  merge  into  one  another  to  form  a  continuum,  and  we 
get  exactly  the  elastic  spectrum  that  we  have  just  been  con- 
sidering. We  see  from  this,  that  the  more  we  are  justified 
in  replacing  the  finite  crystal  by  one  of  infinite  extent  the 
better  our  results  if  we  know  only  the  distribution  law  of  the 
natural  frequencies  (without  knowing  their  position  exactly). 
The  law  of  distribution  of  the  natural  frequencies,  which 
was  discovered  by  Born  and  Kdrmdn  and  extended  by  Born 
in  his  "  Dynamics  of  the  Crystal  Lattice  "  to  the  most  general 
type  of  crystal,  may  be  formulated  thus :  Select  from  the 
totality  of  all  elastic  loaves  the  small  group,  whose  lengths  lie 
between  A.  and  \  +  d\,  and  ivhose  normal  direction  lies  in  the 
elementary  solid  angle 138  dO.  Each  of  the  3s  branches  of  the 

Y 
spectrum  then  contribute   —  dXdto  natural  frequencies  to  this 

group.     Here  V  denotes  the  volume  of  the  finite  crystal. 


§  9.  Continuation.     The  Atomic  Heats  at  Low,  very  Low,  and 
High  Temperatures 

The  knowledge  of  this  law  of  distribution  allows  us  to 
write  down  at  once  the  thermal  capacity  of  the  crystal  con- 
sisting of  Ns  particles.  From  (42)  it  is : 


50 


THE  QUANTUM  THEORY 


.     (52) 


This  formula  is  to  be  interpreted  as  follows  :  the  natural  fre- 
quencies vi  are,  by  (48),  to  be  expressed  as  functions  of  the 
wave-length  X  and  the  wave-direction  n  :  then  the  integra- 
tion is  to  be  performed  with  respect  to  X  from  the  smallest 
wave-length  Xm(w),  which  itself  depends  upon  the  wave- 
direction  n,  up  to  the  maximum  X  =  oo  .  The  result  of  this 
integration  still  depends  on  the  wave-direction  and  the  index 
i.  Finally,  integration  is  to  be  performed  over  all  directions 
(that  is,  over  all  elementary  solid  angles  between  0  and  4?r) 
and  summation  over  all  3s  branches  of  the  spectrum.  But 
we  have  seen  that  the  3s  branches  of  the  spectrum  fall  into 
two  groups.  The  first  3  branches  (i  =  1,  2,  3)  contain  the 
totality  of  slow  acoustic  natural  frequencies;  for  these 
branches  we  have  the  dispersion  law  (49)  which  is  valid  for 
long  waves.  The  remaining  3(s  -  1)  branches  contain  the 
totality  of  the  quick  (infra-red  and  ultra-violet)  natural  fre- 
quencies, with  the  entirely  different  type  of  dispersion  law 
(50),  which  also  holds  for  long  waves.  Hence  the  sug- 

3t 

gestion  naturally  occurs  of  dividing  the  sum  J    of  (52)  into 

i  =  l 

two  parts,  corresponding  to  the  two  different  groups  of  fre- 
quencies and  of  writing 

rp  =  r?  +  r<*> 

where 

JL  i       (53) 

r(>  =  kv      •  •  •  ;      r<*>  =  kv 


These  still  very  complicated   formulae  may,  according  to 
Born,  be  brought  into  a  very  simple  and  comprehensive  form 


LATTICE  THEORY  OF  ATOMIC  HEATS     51 

by  limiting  our  considerations  to  low  temperatures  and  in- 
troducing certain  approximations.  As  we  have  already  re- 
cognised, at  low  temperatures  only  the  long  waves  contribute 
to  the  energy-content.  Hence  we  shall  apply  in  formula  (53) 
all  those  approximations  which  are  introduced  by  confining 
ourselves  to  long  waves.  Let  us  consider  first  r(;p.  Here  we 
set  in  place  of  the  v{S  of  (50)  the  constant  values  v°.,  which 
are  independent  of  the  wave-length  A.  and  of  the  wave- 
direction.  If  we  do  this,  we  can  place  the  constant  factors 


in  front  of  both  integration  signs,  and  write 


!«-*>      V^,.    FU>3    ,  where,, 


The  factor  in  square  brackets  has,  however,  a  simple  meaning. 
From  the  law  of  distribution  of  the  natural  periods  we  see, 
namely,  that  this  factor  gives  the  sum-total  of  all  natural 
frequencies  that  occur  in  one  of  the  3s  branches  of  the 
spectrum  ;  it  therefore  has  the  value  N,  which  as  has  already 
been  said,  is  the  number  of  basic  groups  which  go  to  make 
up  the  crystal.  If  we  choose  the  piece  of  crystal  under  con- 
sideration such  that  its  size  is  so  that  N  is  equal  to  the 
Avogadro  number,  then  if  we  remember  that  Nk  =  R  for 
I*2),  the  expression 
3 


follows.  If  we  compare  this  result  with  (34)  we  see  that  rop 
—  excepting  for  the  missing  factor  3  —  consists  of  3(s  -  1) 
Einstein  functions.  We  write  the  expression  in  the  form 

0 

M         where  ^  =_^          .     (55) 
in  which  the  abbreviation  is  obvious.     The  fact  that,  in  using 


52  THE  QUANTUM  THEORY 

these  approximations,  we  come  across  Einstein  factors,  i.e. 
that  we  encounter  the  "  monochromatic  "  theory,  might  have 
been  anticipated.  For  since  we  treated  the  v/s  here  as  con- 
stants that  are  quite  independent  of  wave-length  and  wave- 
direction,  these  vibrations  represent  processes  which  have 
nothing  to  do  with  the  propagation  of  elastic  waves  in  the 
crystal  as  a  whole  :  and  this  means  that  the  individual 
particles,  uncoupled  as  it  were,  perform  3(s  --  1)  mono- 
chromatic vibrations. 

The  approximate  evaluation  of  the  first  part  rW  is  quite 
different.  For  here  we  have  to  use  for  the  frequencies 
vi»  V2>  V3»  *ne  relations  (49),  which  connect  the  three  acoustic 
natural  frequencies  with  wave-length  and  wave-direction. 
Here  we  have  therefore  to  deal  with  three  real  elastic  oscilla- 
tions, which  are  propagated  in  the  crystal  with  the  three 
different  acoustic  velocities  q-^ri),  q*(ii),  qs(n),  each  of  which 
depends  on  the  direction  (n).  The  crystal  acts  here  as  a 
dynamic  whole,  exactly  as  in  Debye's  point  of  view.  Hence 
we  may  conjecture  that  r(J>  allows  itself  to  be  brought 
into  the  form  of  three  Debye  functions  (45).  The  more 
exact  calculation  confirms  this  supposition,  and  gives  us  1M 


•  (56) 


which,  taking  Debye's  formula  (45)  into   consideration,  we 
may  write  in  the  following  immediately  intelligible  form  : 


I^-IDft          .        .        •     (57) 

i=l 

The  three  magnitudes  Xi  here  play  the  part  of  three  upper 
limits  of  frequency.     Their  values  are 


where  the  three  magnitudes  qt  represent  certain  mean  direc- 
tions of   the  acoustic  velocities,  which  therefore  no  longer 


LATTICE  THEORY  OF  ATOMIC  HEATS     53 

depend  on  the  wave-direction.    From  (55)  and  (57)  we  get  for 
the  thermal  capacity  of  the  piece  of  crystal  considered 


-     (59) 


Now,  since  JV  particles  of  each  of  the  s  different  kinds  of 
particles  are  present,  that  is  one  gramme-atom  of  each  kind 
of  particle  exactly  —  for  N  is  the  Avogadro  number  —  the  piece 
of  crystal  contains  s  gramme-atoms  of  different  sorts  of 
particles.  If,  therefore,  we  cut  the  crystal  into  s  equal 

N 
pieces  in  such  a  manner,  that  each  piece  comprises  only  - 

basic  groups,  then  each  of  these  pieces  contains  a  so-called 

"mean"  gramme-atom.     Hence  if  we  now  consider  only  a 

•p 
single  one  of  these  pieces,  its  thermal  capacity  is  —  ;  we  call 

S 

it  the  "  mean  atomic  heat  "  Cv,  and  we  may  write 
3 
D       +  (*         .        .     (60) 


Here  the  #/s  have  the  same  meaning  as  in  (58).     For  the 

N 

piece  of  crystal  now  under  consideration  consists  of  —  basic 

s 

groups,  and  has  therefore  the  volume   — .      Formula  (58), 
however,  obviously  remains  unchanged  when  we  replace  in 

it  N  and  V  by  —  and  — .     The  quantity  -,  the  volume  of  a 
s  s  s 

mean  gramme-atom,  is  also  called  the  mean  atomic  volume. 

In  the  case  of  chemical  compounds,  in  which  several  sorts 
of  atoms  occur  in  the  basic  group,  and  also  in  the  case  of 
polyatomic  elements,  in  which  the  basic  group  contains  several 
particles  of  a  like  sort,  we  frequently  speak  of  the  molecular 
heat.  In  doing  so,  we  follow  the  usual  chemical  conception, 
inasmuch  as  we  imagine  the  s  particles  of  the  basic  group 
divided  into  one  or  several  sub-groups,  and  regard  each  sub- 
group, taken  alone,  as  a  molecule.  If  then  the  molecule 


54  THE  QUANTUM  THEORY 

contains  q  atoms,  then  qC,,  is  the  mean  molecular  heat;  for 
example,  the  basic  group  of  rock-salt  (NaCl)  contains  one 
sodium  ion  and  one  chlorine  ion.  The  whole  piece  of  crystal, 

which,  by  definition,  contains  -  =  —  basic  groups,  comprises 

S          A 

therefore  —  sodium  ions  and  the  same  number  of  chlorine 
ions,  that  is  to  say  -  "  NaCl-molecules."  q  is  in  this  special 

case  equal  to  2.  Hence  2C,,  represents  the  thermal  capacity 
of  N  "  NaCl-molecules,''  that  is,  the  mean  molecular  heat  of 
rock-salt. 

If  among  the  s  particles  of  the  basic  group  there  are  p 
atomic  residues  and  s  —  p  electrons,  the  number  of  Einstein 
factors  in  (59)  reduces  to  3(p  -  1),  since  the  3(s  -  p)  ultra- 
violet frequencies  arising  from  the  s  -  p  electrons  contribute 
only  in  a  vanishingly  small  degree  to  the  atomic  heat  as  com- 
pared with  the  infra-red.  We  thus  arrive  at  the  law  :  the 
mean  molecular  heat  of  a  crystal  wJiose  basic  group  includes  p 
(similar  or  different)  atomic  residues,  is  made  up,  at  a  suf- 
ficiently low  temperature,  of  three  Debye  terms  (with,  in  general, 
three  different  upper  limits  of  frequency)  and  3{p  -  1)  Einstein 
terms  (in  which  the  3(p-I)  infra-red  natural  frequencies  for 
long  waves  appear  as  frequency  numbers). 

When  we  descend  to  the  lowest  temperatures,  the  Einstein 
terms  disappear  exponentially,  and  only  the  three  Debye  terms 
remain,  for  these,  as  we  know,  decrease  much  more  slowly. 
In  them  we  can  further  replace  all  the  upper  limits  of  the 
three  integrals  (see  (56))  by  GO  ,  so  that  the  integrals  thereby 
become  numerical  constants.  Eemembering  (58)  we  get  the 
fundamental  law,  that  the  molecular  heat  of  every  crystal  at 
the  loivest  temperatures  is  proportional  to  the  third  power  of 
the  absolute  temperature.  So  the  general  lattice  theory  con- 
firms Debye's  result.  The  formula  obtained  has  the  following 
simple  form  :  141 


where  VA  is  the  "  mean  atomic  volume  " 

mean  atomic  weights 

-  _  -  .  -  i*  -    1 


TESTS  OF  THE  BORN-KARMAN  THEORY    55 

and  q  represents  a  quantity  which,  if  suitably  defined,  may  be 
called  the  mean  acoustic  velocity,  introduced  in  place  of  the 
three  different  acoustic  velocities  qlt  q2,  qs. 

Also  in  the  other  extreme  case,  for  high  temperatures,  a 
very  useful  formula  can  be  obtained,  as  H.  ThirringiK 
showed.  He  started  from  (52)  and  developed  the  exponential 
functions  in  series.  The  following  value  is  then  obtained  for 
the  mean  atomic  heat  : 


where  the  coefficients  Jv  J2,  J3,  .  .  .  depend  in  a  complicated 
manner  on  the  elastic  constants  of  the  crystals,  the  atomic 
masses,  and  the  atomic  distances. 

§  10.  Tests  of  the  Born-Karman  Theory 

How  do  matters  stand  with  regard  to  the  testing  of  the 
Born-Kdrmdn  Theory?  We  see  at  once  that  it  is  incom- 
parably more  difficult  than  in  the  case  of  Debye's  Theory  :  for 
even  in  simple  cases,  the  calculation  of  the  mean  atomic 
heat  of  a  crystal  is  very  complicated,  and  requires  above  all 
a  more  exact  knowledge  of  its  elastic  behaviour  than  we  at 
present  possess.  Only  by  restricting  our  attention  to  low 
and  very  low  temperatures  on  the  one  hand,  where  the 
formulae  (60)  and  (61)  may  be  applied,  and,  on  the  other, 
to  the  region  of  high  temperatures,  within  the  limits  of 
applicability  of  Thirring's  formula  (62),  are  we  enabled  to 
carry  our  calculations  for  a  number  of  simple  substances  to 
the  point  of  comparison  with  experimental  results.  Born 
and  Kdrmdn  themselves,  in  one  of  their  first  publications  14a 
tested  the  formula  (61),  valid  for  the  lowest  temperatures 
(Debye's  TMaw),  by  comparing  its  results  with  those  of 
experiment.  They  limited  themselves  in  this  case  to  metals 
(Al,  Cu,  Ag,  Pb)  which,  however  —  at  any  rate  in  the  usual 
form  —  are  not  proper  crystals,  but  irregular  crystalline 
aggregates.  For  this  reason,  they  proceeded  as  if  the  metal 
were  an  isotropic  body,  and  obtained  the  mean  acoustic 
velocity  —  the  only  quantity  in  (61)  which  in  general  requires 


56  THE  QUANTUM  THEORY 

extensive  calculation — from  the  following  relation  which 
holds  for  isotropic  bodies  : 1H 

Q  1          O 

N+! <68) 

Here  ql  and  qt  are  the  velocities  of  propagation  of  the 
longitudinal  and  transverse  elastic  waves,  magnitudes,  there- 
fore, which  may  be  simply  calculated  from  the  two  elastic 
constants  of  the  isotropic  body  and  its  density.1*9  The 
agreement  of  the  values  of  Cv  thus  found  with  the  experi- 
mental data  is,  especially  in  the  case  of  Al  and  Cu  (and  also  Pb), 
quite  good.  A.  Euckenw  has,  however,  pointed  out  rightly, 
that  no  weight  should  be  attached  to  this  agreement.  For 
the  values  of  the  elastic  constants  which  Born  and  Kdrmdn 
used  for  calculating  qt  and  qt  are  those  which  are  correct  at 
the  ordinary  room  temperature.  If  we  take  their  dependence 
on  temperature  into  account,  the  good  agreement  between 
theory  and  experiment  disappears.  Metals  are,  indeed,  not 
isotropic  bodies,  and  hence  it  is  not  permissible  to  use  the 
observable  elastic  constants,  which  depend  upon  temperature, 
in  calculating  q. 

Matters  are  much  more  favourable  in  the  case  of  real 
crystals,  in  which,  as  experiments  by  E.  Madelung  i«  show, 
the  elastic  constants  vary  very  little  with  temperature.  But 
here  the  calculation  of  the  mean  acoustic  velocity  q  gives  rise 
in  general  to  notable  difficulties,1*8  which  may,  however, 
be  cleared  away  in  simple  cases  by  a  very  practical  method 
due  to  L.  Hopf  and  G.  Lechner.™9  Eopf  and  Leclmer  were 
thus  enabled  successfully  to  carry  out  the  calculations  for 
sylvin  (KC1),  rock-salt  (NaCl)  fluor-spar  (CaF2)  and  pyrites 
(FeS2).  They  proceeded  to  calculate  the  quantity  q  from  the 
observed  values  of  Cv,  assuming  the  correctness  of  formula 
(61),  and  they  then  compared  these  with  the  value  of  q 
calculated  from  elastic  data.  The  result  showed  very  satisfac- 
tory agreement.  1»o 

It  is  of  particular  interest  to  test  the  very  clear  formula 
(60)  which  gives  the  mean  atomic  heat  as  a  sum  of  three 
Debi/e,  functions  and  3(s  -  1)  Einstein  functions.  Here  the 
three  infra-red  natural  frequencies  v°,  vj?,  v°  coincide,  and  the 


TESTS  OF  THE  BORN-KlRMAN  THEORY    57 

three  Einstein  functions  become  equal  to  one  another.  If  we 
introduce  the  further  approximation  of  replacing  the  three 
different  quantities  Xi  in  the  Debye  formula  by  a  mean  value 
x,  it  follows  that 

C,  =  \{D(x)  +  E(x)}     .        .        .     (64) 

In  this  we  use  the  value  of  x  deduced  from  formula  (58) 
by  merely  replacing  <?,•  in  it  by  a  mean  value  q,  which  can  be 
calculated  by  the  method  of  Hopf  and  Lechner  just  mentioned. 

j,  o 
x,  on  the  other  hand,  according  to  (54),  =  ^ ,  where  v°  is  the 

infra-red  natural  frequency  of  the  crystal  (for  long  waves), 
which  may  be  determined  from  the  dispersion  in  the  infra- 
red or  by  the  method  of  residual  rays. 

Formula  (64)  had  already  been  given,  previously  to  Born, 
by  W.  Nernst,in  who,  however,  based  his  argument  on  a 
supposition  which  is  no  longer  tenable.  Nernst  started 
from  the  conception  that,  for  example,  in  the  case  of  rock- 
salt,  the  NaCl-molecules  are  located  upon  the  points  of  the 
space -lattice,  and  that  the  most  general  state  of  oscillation 
of  the  lattice  arises  from  the  superposition  of  two  modes  of 
motion,  firstly  the  oscillation  of  the  whole  molecules  in  the 
lattice-structure,  which  give  a  Debye  term,  and  secondly  the 
intra-molecular  oscillations  of  the  two  atoms,  which,  being 
almost  monochromatic,  lead  to  an  Einstein  term.  The 
agreement  of  the  Born-Nernst  formula  (64)  with  the  ex- 
perimental data  is  not  very  satisfactory  in  the  case  of  NaCl 
and  KC1,  but  much  better  in  the  case  of  AgCl,  which  belongs 
to  the  same  crystal  type.182  The  reason  for  this  is  believed 
by  E.  Schrodinger  ***  to  lie  in  the  excessively  rough  ap- 
proximation inherent  in  formula  (64). 

Finally,  Thirring's  formula  (62)  has  also  been  tested,  by 
Thirring  himself, IM  for  NaCl,  KC1,  and,  by  neglecting  certain 
factors,  for  CaF2  and  FeS2.  Taking  into  account  the  variation 
of  the  elastic  constants  with  temperature  (which,  however,  is 
to  be  regarded  as  uncertain  and  provisional  since  the  values 
are  only  obtained  by  interpolation)  he  found  good  agreement 
between  theory  and  experiment.  In  connection  with  the 
Thirring  formula,  Born1**  has  also  calculated  the  atomic 
heat  of  diamond  and  compared  it  with  experiment.  Since  in 


58  THE  QUANTUM  THEORY 

this  case,  however,  the  elastic  constants  were  unknown,  Born 
proceeded  to  evaluate  the  curves  of  atomic  heat  for  various 
possible  values,  and  to  select  from  them  that  curve  which 
conformed  most  closely  to  the  results  of  observations.  Thus, 

for  example,  the  value  O63  x  10  ~12  ,£m—    was  obtained  for 

LdyneJ 

the  compressibility ;  this  is  in  satisfactory  agreement  with  the 
value,  probably  too  small,  measured  by  W.  Bichards,  viz. 

0-5x10-4^1. 
LdyneJ 

From  all  this  we  see  that  the  possibilities  of  testing  the 
Born-Kdrmdn  Theory  of  Atomic  Heats,  partly  on  account  of 
the  great  difficulties  of  calculation,  partly  on  account  of  our 
insufficient  knowledge  of  the  elastic  behaviour  of  crystals,  are 
exceedingly  sparse,  so  that  for  the  present  Debye's  much  more 
tractable  formula  (if  necessary,  with  the  addition  of  Einstein 
terms)  appears  more  useful.  If,  in  spite  of  this  fact,  so  much 
space  has  been  devoted  here  to  the  Born-Kdrmdn  Theory,  the 
reason  is  to  be  sought  in  the  conviction  that  this  theory  has 
gone  much  further  than  that  of  Debye  into  the  kernel  of  the 
matter.  For,  without  a  more  exact  treatment  of  the  structure 
of  the  space-lattice  and  its  dynamics,  our  knowledge  of  the 
nature  of  the  solid  state  must  without  doubt  remain  faulty. 

§  ii.  The  Equation  of  State  of  a  Solid  Body 
Linking  up  with  this  new  development  of  the  theory 
of  atomic  heats,  a  number  of  investigators,  chiefly  E. 
Gruneisen,1*6  S.  Ratnoivski,161  and  P.  Debye,***  have  worked 
out  a  theory  of  the  solid  state  with  the  object  of  creating  as 
a  counterpart  to  the  Kinetic  Theory  of  Gases  a  Kinetic 
Theory  of  Solids.  One  of  the  main  problems  in  this  con- 
nexion is  to  formulate  an  "  Equation  of  State,"  that  is,  a 
relation  between  pressure  (p),  volume  (V),  and  temperature 
(T),  a  problem,  which,  according  to  the  doctrine  of  thermo- 
dynamics, is  to  be  regarded  as  solved  as  soon  as  the  "  free 
energy  "  F  of  the  body  is  known  as  a  function  of  the  tempera- 
ture and  the  volume.189  Then  the  pressure,  for  example,  will 
follow  from  the  simple  equation 


EQUATION  OF  STATE  OF  A  SOLID  BODY     59 

which,  as  a  relation  between  p,  V,  and  T,  gives  the  equation 
of  state  at  once.  If  this  is  known,  we  have  mastered  quanti- 
tatively the  behaviour  of  the  body  for  all  changes  of  state. 
For  example,  the  coefficients  of  expansion  a,  and  the  com- 
pressibility K,  result  from  the  well-known  formulae 

•  •«*» 

(F0  is  the  volume  at  the  zero-point.) 

P.  Debye  16°  was  the  first  to  draw  attention  to  the  fact  that 
the  model  of  the  solid  body  which  forms  the  basis  of  the 
atomic  heat  theories  of  Einstein,  Debye,  and  JBorn-Kdrmdn, 
is  necessarily  too  highly  idealised  ;  for  this  idealised  solid 
body  has,  as  is  easily  seen,  a  zero  coefficient  of  expansion.  In 
fact,  if,  as  has  always  been  assumed  hitherto,  the  forces 
which  pull  the  atoms  back  into  their  position  of  equilibrium 
are  proportional  to  the  first  power  of  their  relative  displace- 
ments (assumption  of  quasi-elasticity,  Hooke's  Law),  then 
the  atoms  will  execute  symmetrical  oscillations  about  this 
position  of  rest.  If  this  supposition,  viz.  Hooke's  Law,  be 
valid  for  all  temperatures,  then  the  mean  volume  of  the 
body — that  is,  the  volume  that  it  possesses  when  all  atoms  are 
exactly  in  their  positions  of  rest — must  be  just  as  often  over- 
shot as  undershot,  however  great  the  amplitude  of  the  heat- 
vibrations  may  be.  Hence,  if  we  warm  the  body  from  zero 
until  it  possesses  the  volume  VQ,  and  if  we  assume  that  all 
atoms  are  at  rest  at  zero,  then  its  mean  volume  at  any  tem- 
perature will  also  be  equal  to  F0.  The  body,  therefore,  does 
not  change  its  mean,  observable  volume  with  rise  of  tempera- 
ture ;  its  coefficient  of  expansion  is  therefore  0.  If  we  desire  to 
represent  the  actual  behaviour  of  the  solid  body,  namely,  its 
expansion  when  heated,  as  known  to  us  from  thousandfold  ex- 
perience, we  are  necessarily  obliged,  according  to  Debye,  to  re- 
place Hooke's  Law  of  Force  by  an  expression  involving  higher 
powers  of  the  variation  of  atomic  distance.  Then  the  oscilla- 
tions of  the  atoms  become  unsymmetrical,  and  there  occurs  a 
displacement  of  their  position  of  rest  as  the  energy  of  vibration 
increases.  If  we  arrange  the  generalisation  of  Hooke's  Law 
so  that  a  greater  force  is  necessary  to  bring  the  atoms  nearer 
together  than  to  separate  them,  then  the  change  in  the 


60  THE  QUANTUM  THEORY 

position  of  rest  occurs  in  such  a  manner  that  for  increasing 
energy  of  vibration,  that  is,  for  rise  of  temperature,  the 
relative  distances  of  the  atoms  increase,  and  hence  the  body 
increases  in  volume.  Debye  has  extended  the  theory  in  this 
sense.  Among  other  things  this  gives  us  the  law  previously  de- 
duced by  Griineisenm  that  at  sufficiently  low  temperatures  the 
thermal  coefficient  of  expansion  a  is  proportional  to  the  specific 
heat.  Moreover,  the  very  small  change  in  compressibility 
with  temperature  is  well  accounted  for  on  Debye 's  Theory. 

§  12.  The  Thermal  Conductivity  of  Solid  Bodies  according  to  Debye 

The  importance  of  Debye' s  Theory  is  by  no  means  confined 
to  thermal  expansion.  On  the  contrary,  it  became  manifest 
that  another  important  group  of  phenomena  require  this 
generalisation  of  Hooke's  Law.  In  the  idealised  solid  body, 
in  which  the  elastic  forces  obey  Hooke's  Law,  the  elastic 
waves  will  become  superposed  without  disturbance,  and  will 
penetrate  the  whole  body  without  becoming  weakened.  If 
we  imagine  the  idealised  body  as  a  horizontal,  infinitely 
extended  plate  of  finite  thickness,  and  if  we  transmit  a 
powerful  motion  (high  temperature)  to  the  upper  layer  of 
atoms,  while  we  keep  the  lower  layer  at  rest  (i.e.  at  zero 
temperature),  then  an  elastic  energy  current  (heat  current) 
will  pass  continually  from  above  to  below.  An  energy 
gradient  (temperature  gradient)  does  not,  however,  exist 
in  the  body,  since,  on  account  of  the  undamped  character 
of  the  wave,  the  mean  density  of  energy  is  everywhere  the 
same.  Since,  in  general,  the  conductivity  for  heat  is  equal 
to  the  flux  of  heat  divided  by  the  gradient  of  temperature, 
it  follows  that  the  idealised  solid'  body  possesses  an  infinite 
thermal  conductivity.  The  case  becomes  different,  however, 
if  we  extend  Hooke's  Law  in  the  manner  described,  and  thus 
pass  over  to  the  "  real "  solid  body.  The  waves  in  the  body 
will  then,  on  account  of  the  departure  of  the  equations  of 
motion  from  linearity,  no  longer  pass  over  one  another  un- 
disturbed. On  the  contrary,  an  oscillation  already  present 
will,  in  consequence  of  the  fluctuations  in  density  caused  by 
it,  disturb  the  oscillations  superimposed  upon  it,  with  the 
effect  that  a  scattering,  and  therefore  a  weakening  of  the 
waves  in  the  body  results,  in  precisely  the  same  way  as  a 


THE  ELECTRON  THEORY  OF  METALS     61 

"  cloudy  "  medium  scatters  and  weakens  light  passing  through 
it.  Hence,  in  the  case  taken,  a  temperature  gradient  is  set 
up  in  the  plate  from  the  top  to  the  bottom.  In  the  case  of 
the  real  body  we  thus  arrive  at  a  finite  thermal  conductivity. 
The  mathematical  development  of  this  conception  led  Debye  to 
the  law 162  that  the  thermal  conductivity  of  crystals  is  inversely 
proportional  to  the,  absolute  temperature  (if  we  confine  ourselves 
to  temperatures  which  are  so  high  that  classical  statistics  are 
applicable).  This  deduction  seems  to  be  in  excellent  agree- 
ment with  experimental  results  obtained  by  A.  Eucken.1*3 

§  13.   The  Electron  Theory  of  Metals  and  its  Modification  by  the 
Quantum  Theory 

If  matters  are  already  complicated  in  the  intrinsically 
clear  case  of  crystals,  the  position  becomes  still  more 
difficult  when  we  turn  to  metals  which,  in  general,  con- 
sist of  an  irregular  conglomerate  of  crystallites.  In  this 
case  the  conductivities,  namely,  of  heat  and  electricity,  are 
particularly  deceptive.  According  to  the  classical  theories  of 
P.  Drudew  E.  Riecke,™  and  H.  A.  Lorentz,™  these  pheno- 
mena are  brought  about  by  the  free  conductivity-electrons, 
which,  like  gas-molecules,  fly  about  in  the  space  between  the 
fixed  atomic  residues,  exchange  energy  with  these  upon 
collision,  and  so  take  part  in  the  establishment  of  thermal 
equilibrium.  Thus  the  conduction  of  electricity  is  explained 
as  follows :  in  a  piece  of  metal  of  uniform  temperature,  an 
equal  number  of  electrons  fly,  on  the  average,  in  each 
direction  through  an  element  of  surface.  Hence,  on  the 
average,  there  is  no  transport  of  electrical  charges  through 
this  element  of  surface,  that  is,  no  electric  current  is  flowing 
in  the  piece  of  metal.  If  now  we  apply  a  potential  difference 
to  the  ends  of  the  metal,  an  electric  field  exists  in  the  metal, 
and  this  field  impresses  upon  the  electrons  during  their  "  free 
paths  "  (i.e.  their  paths  between  two  encounters  with  atoms) 
a  certain  one-sided  additional  velocity  which  is  super- 
imposed upon  the  irregular  heat-motion.  Now,  therefore, 
more  electrons  will  pass  per  second  through  the  element  of 
surface  in  one  direction  than  in  the  other,  and  since  the 
electrons  carry  a  negative  charge,  and  so  move  against  the 
field,  i.e.  in  a  direction  opposite  to  the  field,  we  have  now  an 


62  THE  QUANTUM  THEORY 

electric  current  in  the  metal.  The  mathematical  calculation 
of  this  simple  conception  gives  for  the  electrical  conductivity 
o-  of  the  metal  167 

N^ 
'-2S5       '         '         '         '     (67) 

Here  N  is  the  number  of  electrons  per  unit  of  volume,  e  and 
ra  charge  and  mass  of  the  electrons,  q  their  average  velocity, 
and  I  their  free  path.  If  we  write  the  expression  (67)  in  the 
form 


we  may,  according  to  the  assumptions  of  the  classical  theory, 
replace  the  mean  kinetic  energy  \m(f  of  the  electrons  by  %kT. 
For  since,  as  we  assumed,  the  electrons  take  part  in 
establishing  heat-equilibruim,  the  law  of  equipartition  of 
kinetic  energy  applies  to  their  motion,  and  there  is  thus 
allocated  to  each  of  the  three  degrees  of  freedom  of  the 
electrons  the  energy  ^kT.  In  this  way  we  arrive  at  the 
formula 

.          -          •      (676) 

Analogously,  we  get  from  Drude's  Theory  the  coefficient  of 
thermal  conductivity  168 

.         .         .         .     (68) 


so  that  a  combination  of  the  two  formulae  leads  to  the 
fundamental  relation 

I-f  -T    ....     (69) 

which  is  the  law  of  Wiedemann- Franz  and  of  Lorenz.m  It 
states  that  the  ratio  of  the  thermal  to  the  electrical  conductivity 
has  the  same  value  for  all  pure  metals  and  is  proportional  to 
the  absolute  temperature. 

Thus  all  appeared  in  the  best  of  order.  The  classical 
theory  appeared  here  also  to  have  worked  successfully  and 
the  law  of  equipartition  celebrated  a  triumph.  But  upon 
closer  inspection,  gaps  appeared  in  the  apparently  solid 
theoretical  structure,  and  serious  doubts  arose.  For  if  the 


THE  ELECTRON  THEORY  OF  METALS     63 

free  electrons  really  took  part  in  the  thermal  equilibrium, 
and  hence  claimed  their  full  share,  fNfcT  (per  unit  of 
volume),  in  the  equal  division  of  kinetic  energy,  then  this 
share  of  energy  should  be  plainly  noticeable  in  the  atomic 
heat  of  the  body,  namely,  to  the  extent  of  f  N*&,  where  N* 
denotes  the  number  of  electrons  in  a  gramme-atom.  Such 
an  increase  in  the  atomic  heat  of  the  metals  as  compared 
with  the  non-metals  (which  contain  no,  or  vanishingly  few, 
free  electrons)  has  never  been  observed.  This  difficulty  could 
be  avoided  by  assuming  that  the  number  of  electrons  is  small 
compared  with  the  number  of  atoms  per  unit  volume,  and 
then  their  contribution  to  the  atomic  heat  would  be  relatively 
small.  But  then  we  should  expect  from  (676)  much  smaller 
conductivities  than  experiment  has  disclosed,  unless  we  were 
to  assume  high  values,  that  are  improbable,  for  the  mean  free 
path,  "a 

Further,  H.  A.  Lorentz^1  has  shown,  as  we  have  seen, 
that,  if  the  law  of  equipartition  for  the  motion  of  the  electrons 
is  assumed,  the  metals  would  radiate  in  the  region  of  long 
waves  according  to  Bayleigh's  Law,  whereas  we  have  un- 
questionably to  expect,  especially  at  low  temperatures,  the 
radiation  to  take  place  according  to  Planck's  Law. 

The  calculated  dependence  of  the  conductivity  on  tempera- 
ture can  only  be  made  to  agree  with  experience  by  making 
particular  assumptions  at  high  temperatures,  whereas  no 
assumptions  seem  to  be  able  to  make  calculation  and  obser- 
vation agree  for  low  temperatures.  At  high  temperatures 
the  resistance  of  the  metals  increases  proportionally  to 

the   temperature,  that   is,  o-  decreases  with  ™.     This   can 

only  be  reconciled  with  (676),  if  the  product  Nlq  is  inde- 
pendent of  the  temperature.  If  we  assume  with  /.  J. 
Thomson112  that  N  increases  proportionately  to  *JT,  then, 
since  q  is  likewise  proportional  to  >JT,  I  must  decrease  with 
=f  a  hypothesis  which,  as  we  shall  see,  has  latterly  been 

upheld  by  several  investigators. 

Now,  although  the  agreement  between  theory  and  experi- 
ment could  thus  be  compelled  by  special  assumptions  at  high 
temperatures,  the  region  of  low  temperatures  revealed  itself 


64  THE  QUANTUM  THEORY 

as  the  vulnerable  point  of  the  theory.  For  experiments  by 
H.  Kamerlingli-Onnes m  in  the  laboratory  for  low  tempera- 
tures at  Leyden  had  shown  that  the  resistance  of  metals 
at  very  low  temperatures  (the  experiments  extended  as  far 
down  as  1'6°  abs.)  falls  away  to  a  quite  extraordinary  degree, 
and  practically  disappears  before  the  zero-point  is  reached. 
At  any  rate,  the  resistance  cannot,  as  follows  in  view  of  what 
has  just  been  said  from  formula  (67&),  sink  proportionately 
to  only  the  first  power  of  the  temperature ;  on  the  contrary 
the  fall  is  without  doubt  proportional  to  a  higher  power. 
That  the  Wiedemann- Franz  Law  also  ceases  to  be  valid  in 
this  region,  has  been  proved  by  experiments  of  C.  H.  Lees 174 
and  W.  Meissner.™ 

In  order  to  escape  from  all  these  difficulties  the  quantum 
theory  was  appealed  to,  and  attempts  were  made,  in  the  most 
varied  ways,  to  make  it  harmonise  with  the  existing  theory. 
A  first  attack  was  ventured  by  W.  Nernst 176  and  Kamerlingh- 
Onnes,111  who  gave  for  the  resistance  of  the  metals  empirical 
formulae  which  linked  up  directly  with  the  form  of  Planck's 
energy  equation  (9)  and  which  gave  the  change  in  the  resist- 
ance with  temperature  satisfactorily.  F.  A.  Lindemannm 
and  W.  Wien119  conceived  more  detailed  theories.  Linde- 
mann  accepts  in  his  first  paper  J.  J.  Thomson's  hypothesis, 
according  to  which  N  is  proportional  to  >JT,  and  retains  the 
equipartition  law  for  the  motion  of  the  electrons,  so  that  q 
also  becomes  proportional  to  JT.  Then,  according  to  (676), 

the  variation  of  the  resistance  -  with  the  temperature  depend* 
<r 

entirely  on  the  mean  free  path  I.  But  this  is,  according  to 
well-known  considerations  of  the  theory  of  gases,  the  greater 
the  smaller  the  "  radius  of  action  "  of  the  metallic  atoms ;  for 
the  electrons  can  pursue  greater  paths  freely,  i.e.  without 
collisions,  the  smaller  the  hindrances  set  in  their  path.  The 
novel  part  of  Lindemann's  Theory  is  the  fact  that  he  brings 
the  radius  of  action  of  the  atom  into  relation  with  its  ampli- 
tude of  swing  in  its  heat-motion.  For  it  is  at  once  obvious 
that  the  atoms  in  this  heat-motion  will  cover  a  greater  space 
in  a  given  time,  and  their  sphere  of  action  will  be  the  greater, 
the  larger  their  amplitude  of  oscillation,  i.e.  the  higher  the 
temperature.  Thus  the  mean  free  path  also  becomes  a 


THE  ELECTRON  THEORY  OF  METALS     65 

function  of  the  temperature,  inasmuch  as  it  is  brought  into 
relation  with  the  energy  of  vibration  of  the  atoms.  But,  for 
the  latter  term,  Lindemann  inserts  the  value  given  by  the 
quantum  theory,  and  finds  for  the  resistance  the  formula  18° 

hv  '         l~fc, ' ~  •""  '  *       ('") 

F-I  V^-i 

where  v  denotes  the  frequency  of  the  atoms  (again  the  mono- 
chromatic theory) ;  A  and  B  are  constants.  For  high  tem- 
peratures W  then  becomes  proportional  to  the  temperature 
T;  for  low  temperatures  W  decreases  exponentially  with 

hv 

e  2kT  to  a  constant  value  B2.  With  the  help  of  this  formula, 
Lindemann  succeeds  in  representing  the  observations  quite 
well  (the  formula  contains  two  constants  which  can  be  mani- 
pulated) ;  but,  since  the  law  of  equipartition  has  been  retained 
for  the  electrons,  the  difficulties  of  the  excessive  atomic  heat 
and  of  Rayleigh's  radiation  formula  remain.  Moreover,  this 
theory  is  unable  to  explain  the  departures  from  the  Wiede- 
mann-Franz  Law  at  low  temperatures;  for  the  mean  free 
path  I — the  only  quantity  dependent  on  T  which  occurs  in  a- 
— disappears  entirely  from  the  formula  (69). 

W.  Wien  attacked  the  question  much  more  radically  than 
Lindemann.  In  order  once  and  for  all  to  get  rid  of  the 
contribution  of  the  electrons  to  the  atomic  heat — this  is  the 
weak  point  of  all  theories  which  make  use  of  the  law  of 
equipartition — he  assumed  that  the  electrons  do  not  take 
part  in  the  thermal  equilibrium,  but  possess  a  velocity  q 
which  is  independent  of  the  temperature.  Moreover,  he 
makes  the  number  N  of  the  electrons  per  unit  volume  equal 
for  all  temperatures.  Then,  according  to  (67),  the  variation 

of   -  with  temperature   is   again   determined  only  by  the 

dependence  of  the  mean  free  path  I  on  the  temperature. 
Wien,  in  a  manner  similar  to  that  of  Lindemann,  connects  I 
with  the  energy  of  vibration  of  the  metallic  atoms,  taking, 
however,  the  complete  elastic  spectrum  into  account  accord- 
ing to  Debye.  He  thus  gets  for  the  resistance  the  value 
5 


THE  QUANTUM  THEORY 

vm 

vdv 
W  =  const.  I   A,  ...     (71) 


(•    vdv 
.  I   A* 

3f--\ 


For  high  temperatures  this  formula  gives  W  =  const.  T,  i.e. 
proportionality  with  the  temperature.  For  low  temperatures 
it  follows  that  W  =-  const.  T2,  i.e.  a  parabolic  decrease. 
The  observations  are  very  well  represented  by  Wien's  formula. 
But,  above  all,  the  unsatisfactory  fact  remains  that  this 
method  does  not  lead  us  on  to  a  theory  of  heat-conduction, 
unless  we  make  new  assumptions,  nor  to  the  Wiedemann- 
Franz  Law.  For,  by  the  condition  that  the  motion  of  the 
electrons  takes  place  quite  independently  of  the  temperature, 
Wien  has  taken  away  the  possibility  of  also  ascribing  the 
transport  of  heat  to  the  electrons. 

This  difficulty  arises  again  in  a  more  recent  paper  of 
F.  A.  Lindemann™  in  which,  in  continuation  of  the  con- 
ceptions of  Born  and  Kdrmdn,  the  supposition  is  introduced 
that — just  as  the  atoms  in  a  crystal — the  electrons  in  a  metal 
form  a  lattice.  F.  Haber 181a  has  also  adopted  a  similar  hypo- 
thesis. The  conduction  of  electricity  is  then  explained  by 
supposing  this  electron  lattice  to  move  practically  as  a  rigid 
structure  relatively  to  the  atomic  lattice  and  so  through  the 
metal.  This  model  has  many  advantages.  Since,  in  the 
heat-motion,  in  which  the  electron  lattice  naturally  takes 
part,  the  electronic  vibrations,  on  account  of  their  mass,  are 
extremely  rapid  (high  frequency),  these  vibrations  of  the 
electron,  according  to  Planck's  formula  for  the  energy,  make 
no  appreciable  contribution  to  the  atomic  heat.  In  addition 
the  abnormal  conductivity  (supra-conductivity)  which  has 
been  observed  at  very  low  temperatures  may,  if  we  use 
earlier  considerations  by  /.  Stark,1**  be  explained  without 
difficulty  by  the  conception  that  at  these  very  low  tempera- 
tures at  which  the  atomic  space-lattice  is  practically  at  rest, 
the  electronic  lattice  glides  almost  unimpeded  through  the 
gaps  of  the  atomic  lattice. 

G.  Borelius,1**  in  a  sketch  which  was  recently  published, 
uses  ideas  similar  to  those  of  Lindemann. 
Finally,  we  may  refer  to  a  paper  by  K.  Herzfeld 1M  which, 


THE  ELECTRON  THEORY  OF  METALS     67 

in  contrast  to  the  preceding  investigations,  attacks  the  ques- 
tion from  a  more  phenomenological  point  of  view  without 
making  use  of  a  particular  model.  For  if,  in  the  formulae 
for  o-  and  y,  (67)  and  (68),  we  bring  into  evidence  the  energy 
E  =  $mq2  of  the  electrons  by  writing  the  equations  thus  : 183 


.        .         .     (72) 


we  get 


as  the  expression  which  represents  the  Wiedemann-  Franz 
Law.  Herzfeld  then  shows  that,  if  we  compare  the  result 
with  observation,  the  formula  (73)  can  be  made  to  agree  well 
with  the  actual  measurements  if  we  set  Planck's  expression 

s  -g~  —  for  E.     (The  factor  %  has  been  introduced  because 

P  -\ 

the  energy  of  the  electrons  is  solely  kinetic.)  The  values  for 
v  which  have  to  be  used  stand  in  no  recognisable  relation  to 
the  atomic  frequencies.  A  paper  by  F.  v.  flawer186  works 
along  similar  lines. 

If  we  survey  the  whole  field  of  the  conduction  of  heat  and 
electricity  in  metals  we  recognise  that  here  the  last  word 
has  not  been  spoken,  and  that  a  great  deal  of  hard  work  will 
be  necessary  to  clear  up  finally  the  extraordinarily  com- 
plicated relationshipsi  But  much  would  doubtless  be  gained 
for  the  theory  if  in  future  the  observations,  as  far  as  possible, 
are  no  longer  made  on  crystal  aggregates,  but  on  metal 
crystals  that  are  pure  and  homogeneous. 


CHAPTEE  V 


The  Intrusion  of  Quanta  into  the  Theory  of  Gases 

§  i.  The  Heat  of  Rotation  of  Diatomic  Gases  according  to  the 
Quantum  Theory 

WHILE  the  molecular  theory  of  the  solid  state  thus 
gained  new  nourishment  from  the  doctrine  of  quanta, 
the  kinetic  theory  of  gases  could  no  longer  be  preserved  from 
the  influx  of  the  new  views.  W.  Nernst 187  had  pointed  out 
quite  early  that  quantum-effects  are  to  be  expected  in  the 
rotation  of  di-  and  polyatomic  gas-molecules,  and  also  in  the 

oscillation  of  atoms  in  the 
molecule.  Let  us  take  as 
an  example  the  diatomic 
gas  hydrogen,  the  mole- 
cule of  which  we  may 
picture  provisionally  as  a 
rigid  "  dumb-bell "  (Fig.  6). 
The  knobs  of  the  dumb-bell 
are  the  hydrogen  atoms, 
the  grip  represents  their 
chemical  bond.  Such  a 
molecule  is  known  to 
possess,  besides  its  transla- 
tory  motion  (three  degrees 
of  freedom),  the  possibility 
of  rotating  about  an  axis  at  right  angles  to  the  line  joining  the 
atoms  (two  degrees  of  freedom,  corresponding  to  the  two  axes 
dotted  in  the  figure).  Eotation  about  the  line  joining  the 
atoms  does  not — if  we  accept  Boltzmann's  conception  of  the 
absolutely  rigid  smooth  atom — come  into  play  in  the  ex- 
change of  energy  by  collision  and  hence  in  the  distribution 
of  energy  among  the  separate  degrees  of  freedom :  for  this 
68 


FIG.  6. 


HEAT  OF  ROTATION  OF  DIATOMIC  GASES 

rotation  cannot  be  changed  by  collision.  Considered  fror 
the  new  point  of  view  of  quanta,  which  rejects  "  rigidity  "  and 
"  smoothness  "  as  an  unjustified  idealisation,  the  position  is  as 
follows  :  The  moment  of  inertia  of  the  molecule  relatively  to 
the  line  joining  the  atoms  is  extremely  small  compared  with 
the  moment  of  inertia  about  either  of  the  axes  at  right  angles 
to  this  line.  Buk  it  is  known  that  rotations  which  take  place 
about  axes  with  small  moments  of  inertia  occur  with  much 
greater  rapidity  than  those  about  axes  with  large  moments  of 
inertia  (the  same  energy  having  been  imparted  in  each  case). 
If,  therefore,  we  identify  the  revolutions  per  second  with  fre- 
quencies, and  use  the  Planck  energy  expression  for  the  energy 
of  rotation  (which  is  not  strictly  correct  quantitatively),  a  line 
of  argument  which  has  already  been  frequently  applied  shows 
us  that  the  rotation  about  the  line  joining  the  atoms  possesses 
only  a  vanishingly  small  share  of  the  energy.  For  the  same 
reason  (high  frequencies)  the  degrees  of  freedom,  which 
correspond  to  the  vibration  of  the  atoms  in  the  molecule, 
become  of  importance  only  at  high  temperatures.  As  a  result 

JcT 
of  all  this,  classical  statistics  gives  us  the  value  2  —  =  TcT 

for  the  mean  energy  of  rotation  of  the  hydrogen  molecule ; 
per  gram  me- molecule  it  therefore  becomes  NkT  =  RT. 
Hence  that  part  of  the  molecular  heat  which  arises  from 

rotation  is  equal  to  R,  that  is,  about  T98  ^-l ,  and  it  is  in- 

deg.' 

dependent  of  the  temperature.  In  crass  contradiction  to  this, 
A .  Eucken 188  found  experimentally  that  the  rotation  part  of 
the  molecular  heat  of  hydrogen  has  the  value  R  demanded  by 
the  classical  theory  only  at  high  temperatures.  On  the  other 
hand,  it  gradually  decreases  as  we  pass  to  lower  temperatures, 
and  approaches  asymptotically  the  value  zero  at  the  abso- 
lute zero.  In  the  immediate  neighbourhood  of  absolute  zero, 
hydrogen  behaves  as  a  monatomic  gas.  Eucken's  result  was 
confirmed  by  experiments  conducted  by  K.  Scheel  and  W. 
Heuse,w  who,  however,  measured  the  values  of  the  molec- 
ular heat  only  for  three  temperatures  (92°,  197°,  and  239° 
on  the  absolute  scale).  This  falling  off  of  the  rotational  heat 
is  without  doubt  a  quantum-effect,  similar  to  the  decrease  in 
the  atomic  heat  of  solid  bodies. 


70  THE  QUANTUM  THEORY 

The  first  attempt  to  calculate  this  phenomenon  theoretically, 
is  due  to  A.  Einstein  and  0.  Stern,190  who  proceeded  as 
follows  :  If  /  and  v  are  the  moment  of  inertia  and  the 
number  of  revolutions  per  second  of  the  molecule  respectively, 
then  its  rotational  energy  is 

JSr-£.(arv)»  .  .  .      (74) 

If,  to  simplify  matters,  we  now  suppose  that  all  molecules 
rotate  with  the  same  mean  number  of  revolutions  v  per  second, 
then  we  can  introduce  for  the  corresponding  mean  energy  of 
rotation 

Er  =  ?(2*W     ....     (75) 
the  theoretical  quantum  value 191 

Er  -  -j^—  (according  to  Planck's  first  theory)      (76) 


Er  =   A-  V     +  -     (Planck's  second  theory)    .    (77) 


From  (76)  or  (77),  by  combining  with  (75),  we  obtain  v  as  a 

dP 
function  of  T.     If  finally  we  form  -5—,  and  multiply  by  the 

Avogadro  number  N,  then  we  get  the  share  of  the  energy  of 
rotation  in  the  molecular  heat,  and  we  see  how  it  depends  on 
the  temperature.  It  thus  appeared  that  only  by  using  the 
expression  (77)  for  Er  would  we  be  enabled  to  get  a  satis- 
factory connexion  agreeing  with  Eucken's  measurements,  a 
fact  which  Einstein  and  Stern  used  at  the  time  as  an  argu- 
ment for  the  existence  of  a  zero-point  energy.  It  must,  how- 
ever, be  emphasised  that  this  theory  can  only  be  regarded  as 
a  first  attempt  to  find  general  bearings  and  that  it  does  not 
fulfil  more  rigorous  requirements.  For  the  Planck  energy 
formulae  used,  (76)  or  (77)  are  valid,  as  is  shown  by  their 
genesis,  only  for  configurations  whose  frequency  v  is  a  con- 
stant quantity  independent  of  the  temperature.  Here,  on 


HEAT  OF  ROTATION  OF  DIATOMIC  GASES     71 

the  contrary,  we  have  made  use  of  a  mean  speed  of  rotation 
v,  dependent  on  the  temperature. 

P.  Ehrenfest  ««  in  1913  built  up  a  theory  of  the  heat  due  to 
rotation  on  a  stricter  basis.  He  had,  however,  to  confine 
himself  to  configurations  with  one  degree  of  freedom,  that  is, 
to  rotations  of  the  molecule  around  a  fixed  axis,  as  at  that 
time  an  extension  of  the  quantum  theory  to  several  degrees 
of  freedom  had  not  yet  been  worked  out.  The  expression 
thus  obtained  for  the  heat  of  rotation  was  then,  in  order  to 
take  into  account  both  degrees  of  freedom,  simply  multiplied 
by  2,  a  method  which  readily  suggests  itself,  but  is  not  justi- 
fiable. Ehrenfest  started,  in  his  calculation,  from  the  original 
form  of  the  quantum  hypothesis,  according  to  which  the 
energy  of  linear  oscillators  may  only  be  whole  multiples  of  hv, 
and  accordingly  made  the  condition  that  the  rotational  energy 
of  a  configuration  with  one  degree  of  freedom  (fixed  axis)  may 

only  consist  of  whole  multiples  of  75-.     The  factor  -£  appears, 

because  the  energy  of  rotation  —  in  contrast  with  the  vibra- 
tional  energy  of  the  oscillator  —  is  solely  kinetic  by  nature. 
The  Ehrenfest  condition  is,  therefore,  according  to  (74)  : 


(n-0,1,2,3  .  .  .)      .     (78) 
hence 

vn  =  £^j  (n  =  0,  1,  2,  3  .  .  .)      .    (79) 

and  by  substitution  in  (78) 

(n  =  0,  1,  2,  3  .  .  .)      .     (80) 


Hence  tJie  molecules  can  only  rotate  with  quite  definite,  discrete 
speeds  vll}  and  correspondingly  acquire  only  a  series  of  discrete 
rotational  energies  E^\  quite  in  agreement  with  the  sense  of 
Planck's  quantum  theory.  It  is  noteworthy  that  these  dis- 
crete rotational  energies  are  related  to  one  another  as  the 
squares  of  the  whole  numbers,  whereas  the  energies  of  the 
Planck  oscillators  are  proportional  to  the  whole  numbers 
themselves.  With  the  discovery  of  the  discrete  values  (80) 
of  the  energy,  the  dynamical  part  of  the  problem  was  solved. 


72  THE  QUANTUM  THEORY 

But  we  require  the  mean  energy  Er  of  a  totality  of  N  similar 
molecules.  It  is  here,  then,  that  the  second,  statistical  part 
of  the  calculation  begins.  If  wn  denotes  the  probability  that 
a  molecule  possesses  the  rotational  energy  E(nJ  at  the  tempera- 
ture T  (wn  is  therefore  the  "distribution-function  "  which  has 
been  extended  in  accordance  with  the  quantum  theory),  then 
the  mean  rotational  energy  of  a  molecule  is  known  to  be  equal 


^  '  wn.     Multiplication  by  N  and  differentiation  with 

n  =  0 

respect  to  T  give  us  immediately  the  heat  due  to  rotation.193 
Ehrenfest  thus  obtained  for  the  relationship  between  the 
rotational  heat  and  the"Hemperature  a  curve  which  could,  it  is 
true,  be  made  to  agree  well  with  the  measurements  obtained 
at  low  temperatures  by  choosing  the  arbitrary  constant  /  (the 
moment  of  inertia)  suitably,  but  at  high  temperatures  it  showed, 
before  reaching  the  classical  value  B,  a  maximum  and  a  sub- 
sequent minimum,  which  did  not  correspond  with  the  existing 
observations. 

We  may  note  here  an  important  consequence  of  equation 
(79),  since  it  has  played  a  noteworthy  r6le  in  the  further 
development  of  the  quantum  theory.  If,  namely,  we  write 
down  the  angular  momentum  (the  moment  of  momentum  194) 
of  the  molecule,  that  is,  the  quantity  p  =  J  x  I2irv,  then  it 
follows  from  (79)  that  only  the  special  quantum  values 

Pn=^       (n-0,1,2,3  .  .  .)   •        .     (81) 

of  the  turning-moment  exist.  This  relation  may  also  be 
deduced  directly  from  the  theory  of  the  quantum  of  action  as 
formulated  in  (30).  For,  if  we  select  as  our  general  co- 
ordinate, in  this  case  q,  the  angle  of  rotation  <f>,  then  the 
corresponding  momentum  or  impulse  p  is  known  to  be 
none  other  than  the  moment  of  momentum.193  It  follows 
from  this,  since  p  is  independent  of  </>,  that 


2irp*  =  rih      .        .        .    (82) 
in  agreement  with  (81). 


HEAT  OF  ROTATION  OF  DIATOMIC  GASES     IS 

In  the  same  way,  on  the  basis  of  Planck's  first  theory, 
namely,  the  conception  that  the  special  quantum  rates  of 
rotation  vn  are  the  only  possible  ones,  and  using  the  dumb-bell 
model,  the  author 196  has  recently  carried  out  the  strict  calcula- 
tion for  structures  with  two  degrees  of  freedom  (free  axes  of 
rotation),  making  use  of  the  later  ideas  of  the  quantum  theory. 
This  stricter  method  likewise  gives  us  curves  for  the  rotational 
heat  which  are  useless,  for  they  also  have  a  maximum  and 
a  subsequent  minimum,  as  in  Ehrenfest's  case.  Only  by 
making  special  subsidiary  assumptions,  such  as  excluding 
certain  quantum  states,  can  we  get  curves  which  rise  steadily 
with  increasing  temperature,  and  which  agree,  at  least  to  a 
certain  extent,  with  observation.197  .  "*  . 


f-r   E,  Er  E'r  Er 

FIG.  7. 

Not  much  more  satisfactory  results  were  obtained  in  those 
investigations  which,  again  with  the  use  of  the  dumb-bell 
model,  were  based  on  Planck's  second  theory.  According  to 
this  theory,  the  discrete  values  vn  of  the  rotational  speeds  are 
not  the  only  possible  ones ;  on  the  contrary,  the  molecule  can 
rotate  with  all  rotational  speeds  between  0  and  oo  ,  and  hence 
can  assume  all  values  of  rotational  energy  between  0  and  oo, 
exactly  like  the  Planck  oscillators  in  Planck's  second  theory. 
The  peculiarity  of  the  special  quantum  values  (80)  for  the 
energy  here  consists  in  the  following  :  imagine  the  energies 
Er  plotted  as  abscissae  (Fig.  7)  and  the  corresponding  prob- 
abilities w  as  ordinates ;  then  a  step-ladder  curve  results, 


74  THE  QUANTUM  THEORY 

the  steps  of  which  lie  exactly  at  the  values  Ety.  The  prob- 
ability that  a  given  value  Er  of  the  rotational  energy  will 
appear  is  therefore  constant  within  the  range  of  energy  between 
E(r?  and  JB,"*1*  but  changes  suddenly  at  the  ends  of  this  range. 
According  to  Planck's  first  theory,  which  allows  only  she 
quantum  values  E%\  the  encircled  points  alone  have  a 
meaning.  Only  at  those  points  is  the  probability  other  taan 
zero,  while  all  intermediate  values  of  the  energy  possess  the 
probability  zero,  that  is,  do  not  occur. 

In  this  case,  too,  the  problem  was  first  solved  for  one 
degree  of  freedom  (fixed  axis  of  rotation).  E.  HolmlK  and 
J.  v.  Weyssenhoff199  found,  in  agreement  with  one  another,  a 
steadily  rising  curve  for  the  rotational  heat,  which  fitted  the 
observations  well  at  low  temperatures,  but  undoubtedly  went 
too  high  at  higher  temperatures  (from  about  140°  abs.  up- 
wards). 

But  when  the  modern  development  of  the  quantum  hypo- 
thesis for  several  degrees  of  freedom,  to  which  we  shall  be 
introduced  later,  was  available,  a  stricter  calculation  for 
free  axes  of  rotation,  i.e.  for  two  degrees  of  freedom,  could  be 
carried  out.  This  problem  was  attacked  on  the  one  hand  by 
M.  Planck,*10  on  the  other  by  Frau  S.  Rotszayn™  but  was 
treated  differently  in  each  case.  Planck  started  with  the 
premise  that  this  problem  belongs  to  the  category  of  so-called 
"  degenerate  "  problems.  This  term  is  to  convey  the  follow- 
ing: the  molecule  rotates,  when  no  external  forces  act 
on  it,  according  to  the  doctrines  of  mechanics,  with  con- 
stant speed  in  a  spatially-fixed  plane.  The  position  of 
this  plane  in  space  must,  so  Planck  argues,  be  of  no  im- 
portance for  the  statistical  state  of  the  molecule.  Hence  the 
condition  of  rotation  of  the  molecule  in  the  sense  of  the 
quantum  theory  is  determined  by  a  single  quantity,  namely, 
the  rotational  energy.  In  spite  of  the  fact,  therefore,  that 
the  problem  is  originally  and  naturally  a  problem  of  two 
degrees  of  freedom — for  the  position  of  the  molecule  in  space 
is  determined  by  two  angles — we  must,  according  to  Planck, 
treat  it  in  the  quantum  theory  as  a  problem  of  only  one 
degree  of  freedom.  The  two  degrees  of  freedom  coalesce,  as 
it  were ;  they  are  " coherent" 

In  contrast  to  this,  Frau  Rotszayn  proceeds  to  turn  the 


HEAT  OF  ROTATION  OF  DIATOMIC  GASES    75 

problem  into  a  non-degenerate  one  by  the  addition  of  an  ex- 
ternal field,  and  after  solving  this  problem,  reduces  the  field 
of  force  till  it  vanishes.  This  method  which  was  also  used 
by  the  author  in  the  paper  above  cited,  appears  to  be  par- 
ticularly advantageous,  when  the  calculation  is  based  on 
Planck's  first  theory,  for  peculiar  difficulties  arise  in  "  degen- 
erate "  cases.  Success  here  decides  in  favour  of  the  second 
method.  For  while  Planck  finds  a  curve  202  which  rises  above 
the  classical  value  to  a  maximum,  and  then  descends  asymp- 
totically towards  the  value  R — and  is  therefore  of  no  use — 
the  calculation  of  Frau  Rotszayn  gives  a  steadily  rising  curve, 
which  agrees  well  with  the  measurements  for  lower  and  higher 
temperatures ;  only  the  value  observed  at  T  =  197°  abs.  lies 
about  10  per  cent  too  low.203 


While  all  the  above-mentioned  investigations  are  based  on 
the  dumb-bell  model,  which  can  only  be  regarded  as  a  pro- 
visional, schematic  construction,  P.  S.  Epstein™*  in  1916 
carried  out  the  corresponding  calculations  for  another  mole- 
cular model  proposed  by  N.  Bohr.w*  This  model  of  the 
hydrogen  molecule,  to  which  we  shall  return  later,  is  built  up 
of  two  positive  hydrogen  atoms,  each  of  which  carry  a  single 
positive  charge,  and  around  the  connecting  line  of  which  two 
electrons,  diametrically  opposite,  rotate  in  a  fixed  circle  at  a 
fixed  rate  (see  Fig.  8).  Since  the  equilibrium  in  this  purely 
electrical  system  is  determined  by  the  play  of  the  Coulomb 
attractions  and  the  centrifugal  forces,  and  since  the  radius  of 
the  electron  is  determined  by  a  quantum  condition,  this  model 
possesses  the  advantage  that  all  its  dimensions  are  completely 


76  THE  QUANTUM  THEORY 

fixed,  so  that  there  is  no  longer  any  question  of  the  arbitrari- 
ness of  the  moment  of  inertia.  The  "  dumb-bell  knobs  "  are 
represented  here  by  the  two  positively-charged  hydrogen 
atoms;  the  rotations  of  the  molecule  hitherto  considered 
would  therefore  correspond  to  those  motions  in  which  the 
molecule  rotates  with  a  moment  of  inertia  J"  about  an  axis 
at  right  angles  to  the  line  joining  the  atoms.  But  to  this 
there  must  very  plainly  be  added  the  rotation  of  the  system 
about  the  axis  of  symmetry  (i.e.  the  line  joining  the  atoms), 
which  results  from  the  extremely  rapid  rotation  of  the  elec- 
trons. The  moment  of  inertia  corresponding  to  this  axis  is, 
in  consequence  of  the  extremely  small  mass  of  the  electrons, 
very  small  compared  with  J.  The  whole  system  obviously 
possesses,  if  we  regard  it  approximately  as  rigid,  the  properties 
of  a  symmetrical  top.  Its  motion  is  therefore,  in  consequence 
of  its  own  rotation  about  the  axis  of  symmetry,  not  a  rotation, 
but  instead  the  well-known  motion,  "regular  precession."206 
Epstein  treated  the  problem  from  this  point  of  view  but  could 
not  obtain  agreement  at  low  temperatures  with  the  moment 
of  inertia  calculated  from  the  model  itself,207  namely,  /  =  2 -82 
x  10  ~ 41.  Presumably,  this  failure  depends  on  the  fact  that 
the  model  does  not  correspond  with  reality,  and  in  fact  we 
shall  see  later,  that  well-founded  doubts  have  arisen  as  to  the 
correctness  of  the  Bohr  model.  We  must  therefore  admit, 
unfortunately  as  one  of  a  number  of  instances  in  the  quantum 
theory,  that  the  important  problem  of  the  rotational  heat  of 
hydrogen  still  awaits  solution. 

§  2.  The  Bjerrum  Infra-red  Rotation-spectrum 

N.  BjerrumM*  has  applied  the  relation  (79)  in  a  very 
interesting  manner  to  the  infra-red  absorption  of  polyatomic 
gaseous  compounds.  These  gases  (for  example  HC1,  HBr, 
CO,  H20  in  the  form  of  steam,  but  on  the  other  hand  not 
the  elementary  gases  H2,  02,  N2,  C12)  show,  according  to  the 
investigations  of  S.  P.  Langley,™  F,  Paschen,™  H.  Eubens,211 
H.  Bubens  and  E.  Aschkinass,™  H.  Eubens  and  G.  Hettner,™ 
W.  Burmeisterpu  Eva  v.  Bahr,*u  extensive  absorption  bands 
in  the  short-  and  long- wave  infra-red.  While  in  the  long- 
wave infra-red  we  account  for  the  absorption  by  the  rotating 
molecule,  which,  composed  of  positively  and  negatively 


INFRA-RED  ROTATION-SPECTRUM          77 

charged  atoms,  act  like  electric  double  poles  and  hence  in 
turning  emit  and  absorb  radiation,  Bjerrum  was  the  first  to 
point  out  that  the  molecular  rotation  must  also  make  itself 
noticeable  in  the  short-wave  infra-red.  For  if  there  exists 
in  this  region  a  linear  vibration  VQ  of  the  ions  in  the  molecule 
relatively  to  one  another  —  and  hence  an  absorption  at  this 
point  —  and  if,  in  addition,  the  whole  molecule  rotates  at  the 
speed  vr,  then  it  is  known  that  there  will  be  produced  as  a 
result  of  the  composition  of  the  vibration  with  the  rotation  218 
two  new  vibrations  (and,  correspondingly,  two  new  regions 
of  absorption)  having  the  periods  ve  +  vr  and  VQ  -  vr,  sym- 
metrically disposed  on  both  sides  of  the  ionic  vibration  VQ. 
On  the  whole,  then,  we  have  three  points  of  absorption  : 
vr,  VQ  ~  VT,  vo  +  vr,  to  which  we  must  add  the  non-rotational 
state  VQ  as  a  fourth.  But  if  now,  according  to  Planck's  first 
theory,  the  molecule  can  only  rotate  with  discrete  speeds  of 
rotation  vn  [see  (79)],  we  get  symmetrically  to  the  original 
position  of  absorption  v  =  VQ  and,  on  both  sides  of  it,  a  series 
of  further  discrete  equidistant  positions  of  absorption  : 

v  =  i/o  +  vn  =  v0  +  n~-—} 

*      \(n  -  I,  2,  3  .  .  .)     .     (83) 
v  =  v0  -  vn  =  v0  -  n 


These  discrete  equidistant  positions  of  absorption  have 
actually  been  found  by  Eva  v.  Bahr  in  the  case  of  water 
vapour  and  gaseous  hydrochloric  acid,  and  were  measured 
later  with  still  greater  accuracy  by  H.  Rubens  and  G.  Hettner 
for  water  vapour.  In  an  examination  carried  out  on  an 
extensive  scale  E.  S.  Imeszl1  has  once  more  thoroughly 
investigated  the  hydrogen  halides  (HC1,  HBr,  HF)  and  con- 
firmed the  law  (83)  for  the  position  of  the  absorption  lines. 
It  was  thereby  found  that  the  middle  line  VQ  was  always 
missing.  From  the  standpoint  of  the  theory  here  described 
this  would  mean  that  the  non-rotational  state  does  not  exist, 
that  is,  that  the  molecules  always  rotate  (zero-point  rotation). 
A.  Eucken?1*  who  discussed  the  results  of  E,  v.  Bahr, 
which  were  at  that  time  the  only  ones  known,  deduced  from 
the  good  agreement  between  observation  and  calculation  that 
Planck's  second  theory  is  not  valid,  for  the  experiments 


78  THE  QUANTUM  THEORY 

seemed  so  obviously  to  prove  that  the  molecule  can  actually 
only  rotate  with  the  discrete  speeds  vn.  This  conclusion, 
however,  is  not  inevitable,  as  Planck  219  showed  in  a  pene- 
trating investigation.  On  the  contrary,  the  observations 
may  after  all  be  explained,  surprising  as  it  may  seem,  on  the 
basis  of  his  second  theory  (continuous  "  classical "  absorp- 
tion ;  all  speeds  of  rotation  possible).  This  curious  result  is 
explained  as  follows :  Let  w(Er)dEr  be  the  probability  that 
a  molecule  possesses  exactly  the  rotational  energy  Er ;  hence 
for  N  molecules  Nw(Er)dEr  will  be  the  number  that  will 
possess  exactly  the  rotational  energy  Er.  These  molecules 
rotate  therefore — according  to  (78) — with  the  speed 

_  1     l2Er 

•""    \       </ 

The  quantity  w(Er)  is  here,  according  to  Planck's  second 
theory,  the  step-ladder  curve  pictured  in  Fig.  7.  Planck's 
calculation  then  leads  to  the  following  result :  the  absorption 
of  an  external  radiation  of  frequency  v*  is  not — as  one  should 
expect — -proportional  to  the  number  of  molecules  having  a 
rotational  speed  vr  =  v*,  that  is,  to  the  quantity  w(Er)  but  to 


its  differential  coefficient  .    This  differential  coefficient 

dEr 

is,  however,  as  Fig.  7  shows,  everywhere  equal  to  zero, 
excepting  at  the  special  quantum  energy-values  E^\  that  is, 
at  the  rotational  speeds  vn.  It  follows  from  this,  that  here 
also,  from  the  standpoint  of  Planck's  second  theory,  absorption 
takes  place  only  at  the  special  quantum  rates  of  rotation  v». 
It  thus  comes  about  that,  at  present  at  any  rate,  the  infra- 
red absorption  spectra  of  the  polyatomic  gases,  contrary  to 
all  expectation,  do  not  decide  one  of  the  most  fundamental 
questions  of  the  whole  quantum  theory,  whether,  namely, 
Planck's  first  or  second  theory  is  correct.  An  important 
remark  must  be  added  here.  The  deductions  of  the  relation 
which  gives  the  position  of  the  infra-red  absorption  bands  is 
half  in  accordance  with  the  classical  and  half  in  accordance 
with  the  quantum  theory.  For  although  the  rotational 
speeds  vn  are  determined  by  the  quantum  theory,  the  resolu- 
tion of  the  oscillation  v0  into  the  two  components  VQ  ±  vn 
are  determined  by  the  classical  methods.  How  to  attack  this 


THE  DEGENERATION  OF  GASES  79 

problem  from  a  point  of  view  entirely  consistent  with  the 
quantum  theory  will  be  seen  later  in  Chapter  VIII. 

§  3.  The  Degeneration  of  Gases 

The  phenomena  described  above  which  were  observed  in 
the  case  of  polyatomic  gases  (falling-off  of  the  molecular  heat, 
and  infra-red  absorption)  justify  fully  the  application  of  the 
quantum  theory  to  motions  of  rotation.  On  the  other  hand, 
the  attempts  to  go  a  step  farther  and  to  apply  it  to  the 
translational  energy  of  gases  rest  upon  a  much  more  insecure 
basis.  If  this  step  is  taken,  the  hitherto  exceptional  position 
occupied  by  the  monatomic  gases,  whose  molecules  contain 
only  translational  energy,  becomes  destroyed,  for  then  they, 
too,  must  succumb  to  the  quantum  law.  This  problem 
has  been  attacked  from  various  quarters  [0.  Sackur,®0  H, 
Tetrode  m  W.  H.  Keesom,™  W.  Lenz  and  A.  Sommerfeld,*** 
P.  Scherrer  ,™  M.  Planck.&S]  Thus,  for  example,  Tetrode, 
Keesom,  Lenz  and  Sommerfeld  imagine  the  thermal  motion 
of  the  gas  split  up  into  a  spectrum  of  natural  frequencies,  and 
they  then  distribute  the  energy  in  quanta,  that  is,  according 
to  formula  (9),  over  the  individual  natural  frequencies,  quite 
analogously  to  the  manner  of  Debye  and  Born-Kdrmdn  in 
the  case  of  solid  bodies.  Scherrer  and  Planck,  on  the  other 
hand,  apply  the  quantum  hypothesis  directly  to  the  motion 
of  the  individual  gas-atoms,  basing  their  argument  on  the 
modern  formulation  of  the  quantum  conditions  for  several 
degrees  of  freedom.  How  such  a  quantum  resolution  of  the 
translator^  motion  is  effected,  is  perhaps  most  easily  seen 
by  the  following  simple  example  (Scherrer)  :  Let  a  gas-atom 
of  mass  ra  fly  to  and  fro  in  a  cube-shaped  space  of  side  a 
with  the  speed  v  parallel  to  one  of  the  edges.  It  then 

executes  a  sort  of  oscillation  with  the  period  "  =  o~-   If  we 
set  its  kinetic  energy,  E  =  %mv2,  according  to  Planck's  first 


theory    =  n-    (n  =  0,  1,  2,  3  .  .  .)  then  it  follows  that 

h     v 


80  THE  QUANTUM  THEORY 

hence 


Hence  the  velocity  of  the  atom  and  its  translatory  energy 
can  acquire  only  discrete,  quantum-determined  values. 

The  calculations  of  the  above-named  investigators  lead  to 
two  important  main  results,  at  least  in  qualitative  agreement  ; 
in  the  first  place,  there  results  an  alteration  in  the  gas  laws 
at  very  low  temperatures.  The  necessity  for  this  "  degenera- 
tion "  of  the  monatomic  gases  had  already  been  recognised  by 
Nernst,  who  deduced  it  on  the  basis  of  his  new  heat  theorem.226 
For  if  the  equation  of  state  of  ideal  gases 


=  W 


p  =  pressure 

V  =  volume  of  a  gramme-atom 


•IT-          -prril     '  .v^^-no  vi  u,  Qiuimmv-aiw^    i  ,Q_, 

J\R  =  absolute  gas-constant 
temperature 

were  exactly  true  for  all  temperatures  down  to  the  lowest, 
then  the  maximum  work  A,  which  could  be  gained  from 
the  isothermal  expansion  of  the  gas  from  the  volume  Vl  to 
the  volume  F2,  would  have,  as  we  know,  for  all  temperatures 
the  value 

v  v 

'  ' 


=  RT  log,  (pj). 


A  = 


For  all  temperatures  down  to  absolute  zero,  -Tm  =  R  log  (  -^ 

would  differ  from  zero,  in  direct  contradiction  to  the  condition 
(38)  of  Nernst'  s  Theorem.  Hence  it  follows  that  in  the  region 
of  the  lowest  temperatures,  the  equation  of  state  (85)  must 
undergo  modification.  In  fact,  experiments  of  0.  SacJcurW 
on  hydrogen  and  helium  appear  to  speak  in  favour  of  the 
existence  of  this  "  degeneration." 

§  4.  The  Chemical  Constants  of  Monatomic  Gases 

The  second  main  result  given  by  the  application  of  the 
quantum  theory  to  monatomic  gases,  is  an  extremely  in- 
teresting relation  of  the  Planck  constant  h  to  the  so-called 


CONSTANTS  OF  MONATOMIC  GASES        81 

"chemical  constant"  of  the  gas,  a  quantity  which  plays 
an  important  part  in  changes  of  the  state  of  aggregation 
(vaporisation,  sublimation)  and  in  chemical  states  of  equi- 
librium. But  it  is  here  specially  emphasised  that  the  re- 
lationship just  mentioned  is  not  bound  to  the  undeniably 
hypothetical  resolution  of  the  translatory  energy  into  quanta. 
On  the  contrary,  0.  Stern  228  has  succeeded  in  deducing  it  un- 
objectionably,  without  applying  the  quantum  tJieory  to  the  gas. 
The  original  method,  which  Stern  adopts,  may  be  shortly 
sketched  here.  Consider  the  process  of  sublimation,  i.e.  the 
passage  from  the  solid  into  the  vapour  state.  Let  the 
vapour  obey  the  gas  laws,  and  let  its  density  be  negligible 
compared  with  that  of  the  condensed  solid.  Then  classical 
thermodynamics  gives  for  the  pressure  p  of  the  saturated 
vapour  as  a  function  of  the  temperature  the  following 
equation  : 

~     +  f  log  T  ~        T  +  c  •  (86) 


Here  X0  is  the  heat  of  vaporisation  (per  gramme-atom)  at 
absolute  zero,  E^  is  the  energy  of  the  condensed  solid  (per 
gramme-atom)  at  the  temperature  T;  the  constant  C,  which 
is  the  chemical  constant  of  the  vapourising  substance, 
remains  undetermined,  according  to  thermodynamics.  On 
the  other  hand,  the  integral  on  the  right-hand  side  of 
equation  (86),  which  contains  the  energy  of  the  solid 
material,  may  be  completely  calculated  upon  the  basis  of 
our  assured  knowledge  of  the  energy-  content  of  the  solids. 
We  only  require  to  assume  the  solid  to  be  a  Born-Kdrmdn 
crystal,  and  hence  to  use  the  quantum-theoretical  value  (41) 
for  E(T\  If  we  now  restrict  ourselves  to  high  temperatures, 
to  a  region,  therefore,  in  which  the  classical  theory  is  valid, 
(86)  assumes  the  form 


log  p  =  -----  S±r-  -  i  log  T  +  3  log^|)  +  C      (87) 
6 


82  THE  QUANTUM  THEORY 

(The  BN  quantities  v;  here  form  the  elastic  spectrum  of  the 
solid  body  ;  v  is  their  geometric  mean.)  The  formulation  of 
this  equation  constitutes  the  first  step  of  Stern's  deduction. 
It  gives  the  result  of  thermodynamics,  extended  by  the 
application  of  the  quantum  theory  to  the  condensed  sub- 
stance. The  second  step  is  the  formulation,  in  accordance 
with  molecular  theory,  of  a  vapour-pressure  formula  for  high 
temperatures,  in  the  region  therefore  of  classical  statistics. 
Here,  also,  the  Born-Kdrmdn  solid  model  is  used  for  the  con- 
densed substance,  and,  on  the  basis  of  known  laws  of 
probability,  the  number  of  the  atoms  is  calculated  which  are 
in  statistical  equilibrium  in  the  vapour  phase.  In  this  way 
the  density  of  the  vapour,  and  hence,  as  a  result  of  the  gas 
laws,  its  pressure,  are  given.  So  Stern  finds 

log,?  -  -  ^  -  i  log  T  +  log 

Here  m  denotes  the  mass  of  an  atom,  and  X'0  is  the  work 
which  is  necessary  to  bring  N  atoms  (N  is  the  Avogadro 
number)  from  complete  rest  to  the  gaseous  state.  An  un- 
determined constant  naturally  does  not  appear  in  this  formula 
deduced  from  pure  molecular  theory.  For  the  molecular 
model  is  completely  determinate,  and  hence  gives  the  absolute 
value  of  the  vapour  pressure,  not  only  its  temperature  co- 
efficient, as  in  the  case  of  thermodynamics.  A  comparison 
of  (87)  with  (88)  shows,  firstly,  that 

jp         .        •         •     (89) 
i 
and  secondly,  that  the  chemical  constant  C  has  the  value 


.        .        .     (90) 

Eelation  (89)  may  be  interpreted  by  making  the  supposition 
that  the  solid  body  already  possesses  an  energy  amounting  to 
3tr 

}  -FT  at  the  absolute  zero,  that  is,  a  "zero-point  energy,"  to 
i 
which  the  latent  heat  of  vaporisation  A0  must  be  added,  in 


CONSTANTS  OF  MONATOMIC  GASES        88 

order  to  set  the  atoms  completely  free  from  their  union  in  the 
crystal.  Equation  (90)  gives  us  the  solution  of  the  problem 
before  us.  It  gives  the  chemical  constant  of  the  monatomic 
gases  as  a  function  of  the  atomic  mass  and  the  universal 
constants  h  and  k.  Nowhere,  however,  in  the  whole 
deduction — this  should  be  emphasised  once  more — has  the 
quantum  hypothesis  been  applied  to  the  gas  itself. 

In  order  to  make  formula  (90)  available  for  comparison 
with  experiment 229  we  may  expediently  introduce  the  molec- 

7~> 

ular  weight  M  =  mN,  and  set  k  —    >:  then 


C  =  C0  +  a  log  M,  where  C0  =  log      ^7      =  10'17 

If  we  finally  use  the  base  10  instead  of  the  natural  base  e  for 
our  logarithms,  and  measure  the  vapour  pressure  not  in 
absolute  measure  but  in  atmospheres,  we  get  the  chemical 
constant  C'  used  by  Nernst,  which  is  related  to  Sterns  vai-ie 
for  C  thus : 

C'  =  xJL*  =  6-0057 


For  it  we  finally  get  the  simple  expression 

C'  =  C'0  +  4  Iogi0  M,  where  C'0  =  -  1-59  .  (91) 
This  formula  has  been  brilliantly  verified  by  experiment.  The 
hitherto  most  trustworthy  measurements  of  vapour  pressure 
and  chemical  states  of  equilibrium  give  in  the  case  of  hydro- 
gen, argon,  and  mercury  the  values 

-  1-69  ±  0-15,     - 1-65  ±  0-06,     -  1-62  ±  0'03 
We    are  therefore  justified  in    saying   with   Stern  that   the 
expression  (90)  for  the  chemical  constant  of  the  monatomic 
gases   is  theoretically  and  experimentally  one  of  the  best 
founded  results  of  the  Quantum  Theory. 


CHAPTEE  VI 

The  Quantum  Theory  of  the  Optical  Series.  The 
Development  of  the  Quantum  Theory  for  several 
Degrees  of  Freedom  23° 

§i.  The  Thomson  and  the  Rutherford  Atomic  Models 

THE  greatest  advance  since  M.  v.  Lane's  discovery  of  the 
method  of  Eontgen-spectroscopy  for  determining  crystal 
structure  was  made  in  the  realm  of  atomic  theory  in  1913, 
when  the  Danish  physicist  Niels  Bohr  placed  the  atomic 
models  in  the  service  of  the  quantum  theory.  Bohr's  labours 
have  in  their  turn  reacted  on  the  quantum  theory  and  fertil- 
ised it,  and  thus  a  marvellous  abundance  of  notable  successes 
have  been  achieved  in  recent  years  through  the  interaction  be- 
tween the  dynamics  of  the  atom  and  "the  quantum  hypothesis. 
Among  serviceable  atomic  models,  the  one  proposed  by 
J.  J.  Thomson  long  occupied  a  much  favoured  position ;  accord- 
ing to  it,  the  electropositive  part  of  an  atom,  which  constitutes 
the  most  important  part  of  its  mass,  is  supposed  to  be  a 
sphere  of  "atomic  dimensions"  (radius  about  10  ~8  cms.) 
filled  with  a  positive  space  charge  in  the  interior  of  which  the 
negative  parts,  the  electrons,  rest  in  a  stable  position  of  equi- 
librium. This  model  has  the  great  advantage  of  explaining 
on  purely  electrical  grounds  the  possibility  of  "  quasi-elastic- 
ally  bound  "  electrons,  i.e.  such  electrons  as,  being  displaced 
<from  their  position  of  rest,  are  drawn  back  into  it  by  a  force 
ivhich  is  proportional  to  the  displacement.^1  And  it  was  just 
with  the  help  of  such  electrons  that,  as  is  well  known,  P. 
Driide?®  W.  Voigt**  M.  Planck,**  and  H.  A.  Lorentz™ 
succeeded  in  building  up  large  regions  of  theoretical  optics, 
namely,  the  theory  of  dispersion  and  absorption,  and  the 
magneto-optical  effects  (magneto-rotation  and  Zeeman  effect). 
84 


THOMSON  AND  RUTHERFORD  MODELS  85 

Moreover,  the  Thomson  atomic  model  was  able,  by  following 
the  classical  doctrine  of  the  theory  of  electrons,  to  do  what 
must  be  demanded  of  every  serviceable  atomic  model,  viz.  to 
explain  the  emission,  as  a  result  of  the  oscillation  of  its 
electrons,  of  sharp,  i.e.  essentially  monochromatic  "  spectral 
lines,"  the  position  of  which,  on  account  of  the  quasi-elastic 
restoring  force,236  was  independent  of  the  intensity  of  the 
excitation,  that  is,  of  the  energy  of  the  oscillations. 

In  three  important  points,  on  the  other  hand,  the  model 
failed  completely.  In  the  first  place  no  success  at  all,  unless 
with  complicated  and  artificial  hypotheses  invented  ad  hoc, 
attended  efforts  to  deduce  from  Tfwmson's  model  the  formulae 
for  the  optical  series,  for  example,  the  simple  formula  for  the 
Balmer  series  of  hydrogen.237  Secondly,  the  model  could  not 
account  for  the  division  of  the  spectral  lines  in  an  electric 
field  as  observed  and  closely  studied  by  J.  Stark™*  (Stark 
effect),  in  spite  of  the  fact  that  it  had  been  found  most 
valuable,  in  the  hands  of  H,  A.  Lorentz,  for  explaining  and 
calculating  the  Zeeman  effect.239  Thirdly,  it  was  not  in  a 
position  to  explain  the  large  individual  deflections,  sometimes 
exceeding  90°,  which,  according  to  H.  Geiger  and  Marsden,™0 
a-particles  undergo  in  passing  through  thin  metallic  foils. 
For  on  their  way  through  the  metallic  foil,  the  a-particles, 
which  are  known  to  be  doubly  charged  helium  atoms,  come 
into  the  neighbourhood  of  the  metallic  atoms  and  are  more 
or  less  deflected  from  their  straight  paths  by  the  electric  fields 
of  the  atoms.  If,  now,  the  metallic  atoms  were  Thomson 
atoms,  the  electric  field  of  these  atoms  would  attain  its 
greatest  value  at  the  surface  of  the  positive  sphere,  at  a 
distance  therefore  of  about  10  • 8  cms:  from  the  centre  of  the 
atom.  For  from  the  surface  outwards  the  field  decreases, 

according  to  Coulomb's  Law,  with    -    while  it  grows  from  the 

centre  to  the  surface  proportionately  to  r.  Those  a-particles, 
therefore,  which  pass  close  to  the  surface  of  the  positive 
sphere,  must  undergo  the  greatest  deflection.  An  easy  ap- 
proximate calculation  shows,  however,  that  the  field  at  this 
distance  from  the  centre  is  far  from  being  strong  enough  to 
explain  the  great  deflections  which  Geiger  and  Marsden  have 
observed.  This  weighty  reason  led  E.  Rutherford  2*1  to  set  up, 


86  THE  QUANTUM  THEORY 

instead  of  the  Thomson  model,  a  new  one,  which  was  able  to 
explain  the  large  deflections  of  the  a-rays.  According  to  the 
Rutherford  atomic  picture,  the  electropositive  part  of  the 
atom  is  compressed  into  an  extremely  small  space  M2  the  so- 
called  nucleus.  Its  charge  E  consists  in  general  of  z  positive 
elementary  charges  e,  so  that  E  =  ze.  Here  z  is,  according  to 
a  hypothesis  of  van  den  Broek™3  the  atomic  number  of  the 
element,  i.e.  the  number  which  gives  the  position  of  the 
element  in  the  series  of  the  periodic  table.  Thus,  for  example, 
z  =  1  for  hydrogen,  2  for  helium,  3  for  lithium,  and  so  on. 
About  this  nucleus  the  electrons  describe  planetary  paths, 
that  is,  circles  or  Kepler  ellipses  with  the  nucleus  as  focus, 
since  the  electrons  are  attracted  by  it  in  accordance  with 
Coulomb's  Law  (inversely  proportional  to  the  square  of  the 
distance). 

In  the  electrically  neutral  atom  having  the  atomic  number 
z,  e  electrons  circle  round  the  nucleus.  For  example,  the 
neutral  hydrogen  atom  consists  of  a  singly  charged  nucleus 
(E  =  e)  around  which  one  electron  revolves  in  a  circular  or 
elliptic  path. 

That  this  Rutherford  model  is  actually  able  to  explain  the 
cause  of  large  deflections  of  the  a-particles  is  seen  at  once ; 
for  the  field-strength  of  the  nucleus,  in  contrast  to  Thomson's 
model,  increases  strongly  up  to  the  immediate  neighbourhood 
of  the  nucleus,  in  accordance  with  Coulomb's  Law;  hence, 
if  the  positively  charged  a-particles  come  very  close  to  the 
nucleus — that  is,  much  nearer  than  10  ~8  cms. — then  they  are 
exposed  to  the  extremely  powerful  repulsion  of  the  nucleus. 

On  closer  examination,  the  Rutherford  atomic  model  dis- 
appoints us  seriously :  for  the  revolutions  per  second,  v,  of 
the  electrons  depend  on  the  energy  of  the  system.2**  If, 
therefore,  we  suppose,  according  to  the  classical  electron 
theory,  that  an  electron  revolving  at  v  revolutions  sends  out 
an  electromagnetic  radiation  of  frequency  v,  then,  since  the 
system  loses  energy  by  this  radiation,  v  must  diminish  cor- 
respondingly. But  this  means  that  the  atom  is  unable  to  emit 
a  sJiarp,  homogeneous  spectral  line. 

§  2.  Bohr's  Model  of  the  Atom 

It  thus  appears  that  we  are  obliged  to  reject  this  model  at 
the  very  outset.  But  the  history  of  physics  has  decided 


BOHR'S  MODEL  OF  THE  ATOM 


87 


otherwise.  With  deep-sighted  intuition,  Niels  Bohr  saw  the 
possibilities  of  Rutherford's  model  and  brought  it  under  the 
quantum  theory  by  making  three  bold  hypotheses.^  In  the 
first  place,  he  assumed  that  the  electron  (or  electrons)  cannot 
revolve  around  the  nucleus  in  all  paths  possible  according  to 
the  view  of  mechanics,  but  only  in  certain  discrete  orbits 
determined  by  the  quantum  theory.  If  we  restrict  ourselves, 
as  Bohr  did  initially,  to  circular  paths,  then  only  those  paths 
of  an  electron  are  allowable  from  the  view  of  the  quantum 
theory  for  which  the  moment  of  momentum  (angular  mo- 
mentum) of  the  revolving  electron  is  a  whole  multiple  of 

-— ,  in  exact  agreement  with  the  quantum  condition  (81)  or 

(82)  for  the  rotating  molecule. 
This  gives,  in  the  simplest 
case  for  the  quantum  paths  of 
the  electrons,  a  discrete  family 
of  concentric  circles  around 
the  nucleus,  with  radii,  which 
are  related  to  one  another  as 
the  squares  of  the  whole 
numbers  (1  :  4  :  9  :  16  : 
— ).  See  Fig.  9. 
Secondly,  these  "  allow- 
able"  orbits  are  stationary ;  pI(Jj  9. 
they  are  in  a  certain  sense 

stable  states  of  motion.  This  stability  is  gained  by  making 
the  radical  condition  that  the  electron — in  striking  contrast 
with  everything  that  the  classical  theory  has  taught  us — 
shall  not  radiate  when  in  the  stationary  paths.  Since  by 
this  "  decree "  the  loss  of  energy  is  abolished,  the  electron 
can  continually  revolve  in  such  a  "quantum  path."  That 
there  are  such  "non-radiating"  paths  in  the  atom,  is  be- 
yond doubt.  Among  other  things,  the  constancy,  in  time, 
of  the  para-  and  ferro-magnetism  of  bodies,  which  is 
generated  by  revolving  electrons,  speaks  in  favour  of  this 
view.  But  how  electrodynamics  must  be  altered  in  order  to 
guarantee  the  non-radiation  of  the  quantum  paths,  and  only 
of  these,  is  a  question  which  as  yet  remains  unanswered.  As 
we  have  now  abolished  the  "classical"  radiation  of  the 


88  THE  QUANTUM  THEORY 

atom,  the  actually  observed  emission  must  be  accounted  for 
by  a  new  hypothesis.  Here,  again  in  direct  connexion 
with  Planck's  original  quantum  rule,  Bohr's  third  condition 
takes  effect :  when  the  electron  passes  from  one  allowable 
quantum  orbit,  in  which  the  energy  is  W2,  into  another 
allowable  quantum  path  of  energy  Wlt  energy  amounting  to 
Wk  -  W\  is  radiated  in  the  form  of  an  energy-quantum  hv  of 
homogeneous,  monochromatic  radiation.  The  frequency  of 
the  radiation  emitted  is  determined  by  "  Bohr's  Frequency 
Condition : " 

We  can  follow  Einstein2*6  in  imagining  the  passage  from 
the  state  of  higher  energy  to  the  state  of  lower  energy  as  a 
sort  of  radio-active  disintegration,  the  occurrence  of  which  in 
time  is  determined  by  chance.  The  details  of  this  passage  and 
the  release  of  energy  accompanying  it  are,  however,  entirely 
obscure  up  to  the  present. 

§  3.  The  Hydrogen  Type  of  Series  according  to  Bohr's  Atomic 
Model 

However  bold  and  unorthodox  Bohr's  three  hypotheses 
may  have  appeared,  their  success  was  surprising.  If  we  apply 
them  to  a  "hydrogen  type"  of  Rut}ierford-a,tom  in  which 
a  single  electron  revolves  around  a  positive  nucleus  with  a 
z-fold  charge,  we  get247  for  the  frequencies  of  the  spectral 
lines,  which  the  electron  emits  in  passing  from  the  ?ith  to  the 
sth  quantum  path,  the  following  values : 

fe,  m  charge  and  mass~| 
J  of  electrons  I        (93) 

~2 j  \s,  n  whole  numbers    J 


Nz* 


If  we  here  set  z  =  1  (hydrogen),  s  =  2,  n  =  3,  4,  5  .  .  .  we 
get  in  exactly  the  same  form  the  empirical  expression  for  the 
Balmer  series  of  glowing  hydrogen  M8 

(n  =  3,  4,  5  .  .  .)         .     (94) 


THE  HYDROGEN  TYPE  OF  SERIES         89 

For  the  constant  N  which  appears  in  the  empirical  formula, 
the  so-called  Eydberg  number,  Bohr's  Theory  therefore  gives 
the  expression 

N=*^   ....     (95) 

If  we  use  here  the  well-known  values 
e  =  4-774  x  10 -10  (Millikan)         h  =  6'55  x  10  ~27  (Planck) 

—  =  1-77  .  107 
me 

then  it  follows  from  (95)  that 

N  =  3-27  .  101S 

while  the  empirical  Eydberg  number  has  the  value  3-29  .  1015. 
This  striking  agreement  and  the  resolution  of  the  Bydberg 
number  into  universal  constants  is  one  of  the  main  achieve- 
ments of  Bohr's  Theory,249  and  forms  a  strong  argument  for 
its  innate  power.  We  may  say  that,  according  to  Bohr's 
original  theory,  the  individual  lines  of  the  Balmer  series  (Ha, 
Hp,  Hy,  .  .  .)  are  emitted  when  the  electron  jumps  from  the 
3rd,  4th,  5th  ...  orbit  into  the  2nd. 

With  this  statement,  however,  the  achievements  of  formula 
(93)  are  not  exhausted.  For  it  includes,  as  we  easily  see, 
further  spectral  series  of  hydrogen.  Namely,  if  we  set  s  =  1, 
n  =  2,  3,  4  .  .  .  we  get  the  ultra-violet  series  that  was  found 
and  measured  by  Lyman.1®0  If  on  the  other  hand  we  set 
s  =  3,  71  =  4,  5,  6  .  .  .  we  get  the  infra-red  Bergmann  series, 
the  first  two  lines  of  which  were  measured  by  F.  Paschen.**1 

The  element  which  follows  hydrogen  in  the  Periodic  System 
is  helium  (atomic  number  z  =  2).  While,  however,  the  con- 
stitution of  the  neutral  helium  atom  with  its  two  electrons 
is  already  more  complicated — according  to  the  latest  investi- 
gations, the  two  electrons  circle  around  the  nucleus  in  two 
different  orbits — the  simply  ionised  helium  atom,  which  has 
therefore  a  single  positive  charge,  is  entirely  "  of  the  hydrogen 
type ; "  for  it  consists  of  a  doubly-charged  positive  nucleus 
around  which  an  electron  rotates.  The  sole  difference,  as 
compared  with  the  hydrogen  atom,  thus  consists  in  the 
doubling  of  the  nuclear^charge,  0  =  2.  The  series  emitted 


90  THE  QUANTUM  THEORY 

from  the  positive  helium  atom  may  therefore,  according  to 
(93),  be  comprised  in  the  formula 

>-*»(?-*}     •    •     •  <96> 

where  N  is  again  the  Rydberg  number  as  defined  in  (98).  If 
we  here  set  s  =  3,  n  =  4,  5,  6  .  .  .  we  get  the  so-called 
"  principal  series  of  hydrogen  "  which  was  observed  by 
Foivler  2fl2  and  very  recently  measured  with  great  care  by  F. 
Paschen.™3  For  s  =  4,  n  =  5,  6,  7  .  .  .  we  get  the  so-called 
"  second  subsidiary  series  of  hydrogen,"  which  was  observed 
by  Pickering  28*  and  Evans.m  Both  series  were,  before  the 
advent  of  Bohr's  Theory,  falsely  ascribed  to  hydrogen. 

A  new  and  extremely  noteworthy  result  of  Bohr's  Theory  is 
revealed,  if  we  allow  for  the  movement  of  the  nucleus  in  our 
calculations.  For,  in  reality,  the  nucleus  is  not  stationary, 
but  nucleus  and  electron  revolve  about  their  common  centre 
of  gravity.  By  taking  this  fact  into  account  we  are  led  to 
a  slightly  altered  expression  for  the  Rydberg  constant.  In 
place  of  (95)  we  get  the  formula 


in  which  M  denotes  the  mass  of  the  nucleus.  It  follows 
from  this  that  for  different  elements,  for  instance,  hydrogen 
and  helium,  the  Rydberg  constant  differs  somewhat  and  is 
smaller  for  hydrogen  than  for  helium  (since  MH  <  Mffe).  In 
general,  the  value  of  the  Rydberg  constant  increases  with 
increase  of  atomic  weight  tending  towards  a  limiting  value. 
All  this  is  in  perfect  agreement  with  the  results  of  many 
years  of  spectroscopic  research. 

In  the  same  way  as  emission,  absorption  has  a  quantum- 
like  character,  according  to  Bohr's  model.  If  light,  say  of 
the  first  Balmer  line  (Ea),  falls  upon  a  hydrogen  atom,  a 
quantum  hv  of  this  external  Ha  radiation  is  used  to  "raise" 
the  electron  into  the  third  quantum  orbit.  An  amount  of 
energy  hvffa  is  taken  from  the  external  radiation,  that  is,  light 
from  the  line  H*  is  absorbed. 


STRUCTURE  OF  THE  PERIODIC  SYSTEM    91 

§  4.  The  Structure  of  the  Periodic  System 

Even  in  his  earliest  papers  Bohr  endeavoured  to  construct 
for  the  higher  elements  as  well  (Li,  Be,  B,  C,  etc.),  in  con- 
nexion with  the  Periodic  System,  suitable  atomic  models 
with  several  rings  of  electrons,  each  occupied  by  several 
electrons,  in  which,  for  example,  the  well-known  octaves  of 
the  system  are  reproduced  by  a  regular  arrangement  of  the 
external  electrons  which  recurs  at  every  eighth  element, 
while  the  number  of  the  electrons  revolving  in  the  outermost 
ring  is  equal  to  the  valency  of  the  element  in  question. 

W.  Kossel 286  arrived  at  a  similar  structure  of  the  atoms  as 
a  result  of  a  profound  investigation  of  the  formation  of  mole- 
cules from  atoms.  Also,  L.  Vegard,251  A.  Sommerfeld®*  and 
B.  Ladenbnrg  2s9  have  constructed  analogous  atomic  models, 
particularly  taking  into  account  the  well-known  up-and-down 
curve  of  atomic  volumes,  and  using  them  to  explain  other 
periodically  varying  properties  (paramagnetism,  ionic  colour). 
These  considerations,  although  they  are  tending  indisputably 
along  the  right  lines  as  far  as  the  general  principles  are  con- 
cerned, are  not  yet  firmly  established  in  detail. 

§  5.  The  Quantum  Hypothesis  for  Several  Degrees  of  Freedom 

While  the  quantum  hypothesis  in  its  most  primitive  form 
demonstrated  in  this  way  its  innate  power  by  entering  the 
field  of  atomic  dynamics,  it  had,  in  doing  so,  gained  little  as 
far  as  its  own  development  was  concerned.  But  the  fruits  of 
Bohr's  Theory  ripened  more  rapidly  than  could  have  been 
divined.  Already  the  year  1915  brought  a  decisive  develop- 
ment :  almost  simultaneously,  Planck  and  Sommerfeld  inde- 
pendently found  the  solution  of  a  problem  that  had  long  been 
a  burning  question,  namely,  the  extension  of  tlie  quantum 
theory  to  several  degrees  of  freedom.  Sommerfeld^0  retained  a 
close  connexion  with  Bohr's  Theory  in  attacking  the  problem. 
The  first  main  condition  of  this  theory  related  to  the  choice 
of  "  allowable  "  stationary  orbits  among  all  those  mechanically 
possible.  According  to  this,  as  we  saw,  only  those  orbits 
were  allowed  for  which  the  moment  of  momentum  (Impuls- 

moment)  p  is  a  whole  multiple  of     -.     This  may  also  be 


92  THE  QUANTUM  THEORY 

expressed  according  to  (81)  and  (82)  thus  :  among  all  mechan- 
ically possible  paths,  only  those  are  allowable  and  stationary 
for  which  the  pJuise-integral  fulfils  the  condition : 

nh      .  (98) 

In  this  quantum  condition  we  are  to  replace  according  to 
(82)  the  general  co-ordinate  q  by  the  angle  of  rotation  (the 
"  azimuth  ")  <f>,  the  impulse  ^>  by  the  "  impulse  (or  momentum) 
corresponding  to  <f>,"  namely,  p^  (the  moment  of  momentum). 
The  integration  is  thereby  to  be  extended  over  the  whole  range 
of  values  of  the  variable  q,  that  is,  in  the  present  case,  from 
0  to  2*-. 

In  the  case  of  the  original  Bohr  Theory,  which  considers 
only  circular  orbits,  there  naturally  exists  only  a  single 
quantum  condition,  namely,  that  for  the  case  q  =  <£,  since 
the  angle  of  rotation  <f>  is  the  only  variable  of  the  path. 
Matters  are  otherwise,  when  we  reject  the  limitation  to 
circular  orbits,  and  hence  take  .STe^/cr-ellipses  into  account. 
Then  each  point  of  the  path  is  determined  by  two  variables, 
namely,  by  the  distance  r  of  the  electron  from  the  nucleus, 
which  is  at  the  focus  of  the  ellipse,  and  by  the  angle  </>  (the 
"  azimuth  ")  which  r  makes  with  a  fixed  direction  (say  with  the 
straight  line,  which  joins  the  nucleus  to  the  perihelion).  In 
this  case  we  are  presented  with  a  problem  of  two  degrees  of 
freedom,  with  two  generalised  co-ordinates,  r  and  $  (polar 
co-ordinates).  The  simple  extension  of  the  quantum  hypo- 
thesis by  Sommerfeld  now  consists  in  setting  up  in  this  case' 
two  quantum  conditions  of  the  form  (98),  one  for  the  co- 
ordinate <£,  which  agrees  with  the  single  quantum  condition  of 
Bohr's  Theory,  and  a  new  one  for  the  co-ordinate  r,  so  that 
the  selection  of  the  stationary  orbits  is  here  determined  by 
the  two  following  equations : 

nh.  .         .       (99) 


n'h  .        .         .         .     (100) 

n  and  n  are  here  whole  numbers,  p$  and  p,.  are  the  impulses 
(momenta)  corresponding  to  the  co-ordinates  <f>  and  r.261     The 


THE  QUANTUM  HYPOTHESIS  93 

integration  in  (100)  is  to  be  taken  over  the  full  range  of 
values  of  r,  that  is,  from  the  smallest  value  rmin  (perihelion) 
to  the  greatest  value  rmax  (aphelion)  and  back  to  the  smallest 
fmin.  (99)  is  called  the  azimuthal  quantum  condition,  n  being 
the  azimuthal  quantum  number ;  (100)  is  the  radial  quantum 
condition,  ri  the  radial  quantum  number. 

In  a  corresponding  manner  the  extension  may  be  carried 
out  for  more  than  two  degrees  of  freedom.  If  the  system  has 
/  degrees  of  freedom,  and  if  it  is  therefore  characterised  by 
the  /  generalised  co-ordinates  qv  q.^,  qs  .  .  .  and  the  corre- 
sponding impulses  pv  p.2,  p3  .  .  .,  then  the  "  allowable " 
movements  of  the  system  are  limited  by  the  /  quantum 
conditions : 

\Pid(li  =  nih  >   p-A  =  nji,  -  •  -   \Pfdqj-  =  njh  .     (101) 
(nv  n.2  .  .  .  HJ-  are  positive  whole  numbers). 

In  every  one  of  the  /  phase-integrals  the  integration  is  to 
be  performed  over  the  full  range  of  values  of  the  co-ordinate 
in  question. 

A  difficulty,  which  arose  here  from  the  outset,  was  the 
question  as  to  which  co-ordinates  ought  to  be  chosen  for  the 
application  of  the  quantum  rule  (101),  or  whether  the  choice 
is  immaterial.  In  general,  we  may  characterise  a  system  of 
several  degrees  of  freedom  by  various  types  of  co-ordinates ; 
for  instance,  we  may  describe  the  Kepler  movement  of  the 
electron  either  by  polar  co-ordinates  r  and  <f>,  or  by  Cartesian 
co-ordinates  x  and  y.  This  question  is  the  more  urgent, 

when  one  considers  that  the  separate  phase-integrals  Ip^;  do 

not  really  become  constants  for  every  choice  of  co-ordinates, 
as  is  required  by  the  quantum  rule  (101) ,262  P.  S.  Epstein263 
and  K.  Schwarzschild2^  have  solved,  independently  of  one 
another,  this  problem  of  the  "  correct  choice  of  co-ordinates  " 
to  a  certain  extent.  Incidentally,  an  interesting  and  sur- 
prising relation  of  the  quantum  rules  (101)  to  a  long-known 
theorem  of  classical  dynamics  was  revealed,  which  had  been 
propounded  by  Jacobi  and  Hamilton,  and  had  hitherto  been 
successfully  applied  in  celestial  mechanics.  Finally,  quite 
lately,  A.  Einstein,26*  by  modifying  the  expression  (101),  has 


94  THE  QUANTUM  THEORY 

put  forward  a  quantum  hypothesis  which  has  the  advantage 
of  being  independent  of  the  choice  of  co-ordinates.  But  a 
closer  discussion  of  these  abstract  investigations  would  lead 
us  too  far  here. 

The  second  formulation  of  the  quantum  hypothesis  for 
several  degrees  of  freedom  is  due,  as  already  mentioned,  to 
M.  Planck.™*  It  is,  as  it  were,  more  cautious  in  its  nature 
than  the  more  radical  attack  of  Sommerfeld.  Planck,  con- 
tinuing directly  from  the  division  of  the  phase-plane  of  linear 
oscillators  already  discussed,  starts  from  the  so-called  Gibbs 
phase-space  to  deal  with  more  complicated  systems.  For  a 
system  of /degrees  of  freedom,  which  is  characterised  by  the 
co-ordinates  qv  q.2  .  .  .  qy  and  the  impulses  plt  p.2  .  .  .  pf, 
the  Gibbs  phase-space  is  that  2/  dimensional  space,  the  points 
of  which  possess  the  2/1  co-ordinates  q1  .  .  .  p/.  Each  point 
of  the  phase-space  (phase-points)  represents,  therefore,  a 
definite  momentary  state  of  the  system  in  question.  Planck 
now  gives  this  phase-space,  in  exact  analogy  to  the  phase- 
plane,  a  cellular  structure,  by  bringing  into  prominence 
certain  specially  distinguished  boundary  surfaces.  At  the 
same  time  the  size  of  the  cells  is  proportional  to  hf.  The 
points  of  intersection  of  those  boundary  surfaces  then  repre- 
sent the  distinctive  quantum  states  or  phases  of  the  system 
(that  is,  according  to  Planck's  first  theory  the  only  possible, 
the  "allowable"  conditions).  In  contrast  with  Sommerfeld' s 
Theory,  in  which  the  motion  of  a  system  of  /  degrees  of 
freedom  is  always  determined  by  /  quantum  conditions,  in 
Planck's,  under  certain  circumstances,  the  case  may  occur 
that  fewer  quantum  conditions  than  degrees  of  freedom  exist, 
so  that  several  ("  coherent ")  degrees  of  freedom  are  limited 
by  a  single  quantum  condition. 

§  6.  Sommerfeld's  Theory  of  Relativistic  Fine-structure 

That  these  theories  had  found  the  kernel  of  the  matter  was 
soon  to  be  shown  by  applying  them  to  Bohr's  atomic  model. 
According  to  them  from  among  all  the  mechanically  possible 
paths,  which  the  electron  can  describe  about  the  £-fold 
positively  charged  nucleus,  the  allowable,  stationary  paths 
must  be  determined  by  the  two  quantum  conditions  (99)  and 
(100).  This  gives,  in  place  of  the  discrete,  quantised  circles 


RELATIVISTIC  FINE-STRUCTURE  95 

of  Bohr,  discretely  quantised  Kepler  ellipses,  among  which 
also  the  Bohr  circles  are  included,  as  special  cases.  And 
further,  the  ellipses  are  quantum-determined,  both  with  re- 
ference to  their  sizes  (i.e.  to  their  major  axes),  and  to  their 
form  (i.e.  the  relation  of  the  axes  to  one  another),  so  that  here 
every  orbit,  as  compared  with  Bohr,  is  characterised  by  two 
quantum  numbers  n  and  w'.267  In  place  of  formula  (93)  for 
the  hydrogen  type  of  series,  we  get  the  general  formula  : a68 

v  =  Nzf,  _ J_    _         1        "I  (102) 

L(s  +  s'Y       (n  +  nj] 

Here  again  N,  the  Rydberg  constant,  is  given  by  (95),  or 
more  exactly  (the  motion  of  the  nucleus  being  taken  into 
account)  by  (97) ;  s  and  s'  are  the  two  quantum  numbers 
(azimuthal  and  radial)  of  the  final  orbit  of  the  electron ;  n 
and  n'  are  the  quantum  numbers  of  its  initial  orbit.  Since 
also,  as  a  result  of  this  more  complete  view  of  Sommerfeld, 
the  number  of  allowable  orbits  is  greatly  increased,  as  com- 
pared with  those  arising  from  Bohr's  Theory  (owing  to  the 
addition  of  the  ellipses),  the  electrons  have  a  great  many 
more  possibilities  in  passing  from  one  orbit  to  another,  that  is, 
the  chances  of  generating  spectral  lines  are  multiplied.  But 
we  easily  recognise  the  following  fact :  if  we  choose  as  the 
final  orbit  of  the  electron  any  one  of  those  orbits,  for  which 
the  sum  of  the  quantum  numbers  s  +  s'  has  a  definite  value, 
say  s  +  s'  =  2,  and  as  initial  orbit,  any  one  of  those  paths, 
for  which  n  +  n'  has  a  definite  value,  say  n  +  n'  =  3,  then 
all  the  different  transitions  of  the  electrons  from  any  one  of 
these  initial  orbits  to  any  one  of  these  final  orbits  generate 
always  the  same  line  (in  the  case  of  the  figures  above  chosen 
it  will  be  the  first  Balmer  line) ;  for  according  to  (102)  the 
frequency  of  the  line  emitted  depends  only  upon  the  sum 
s  +  s',  and  the  sum  n  +  n',  and  on  the  other  hand  not  on  the 
separate  values  of  s,  s',  n,  n'.  It  would  thus  appear  as  if 
nothing  is  gained  physically  by  Sommerfeld' s  elaboration  of 
the  theory  as  compared  with  Bohr's  original  theory.  How- 
ever, as  Bohr  had  already  pointed  out,  the  calculations  are 
incomplete  in  one  important  respect,  which  become  of  funda- 
mental importance  when  consistently  taken  into  account, 
and  which  represents  the  main  achievement  of  Sommerf eld's 


96  THE  QUANTUM  THEORY 

theory  of  spectral  lines.  Namely,  the  velocities  of  the 
electrons,  which  appear  in  these  problems,  cannot  be  con- 
sidered negligibly  small  compared  with  the  velocity  of  light. 
In  this  case,  however,  we  cannot,  as  we  know,  calculate  by 
the  methods  of  classical  mechanics,  which  regards  the  mass 
of  the  electron  as  constant,  but  must  take  our  stand  upon  the 
theory  of  relativity,  and  hence  take  into  account  the  variations 
of  the  mass  of  the  electron  with  its  speed.  Sommerfeld  com- 
pleted the  calculation  in  this  respect.  The  paths  of  the 
electron  and  the  nucleus  differ,  in  this  refinement  of  the 
theory,  from  the  ordinary  Kepler  ellipse  in  that  the  perihelion 
of  the  orbit  advances  in  the  course  of  time,  and  that  the  path 
loses  its  closed  character.  This  has  the  effect  that  the  energy 
of  the  electron  in  the  stationary  quantum-chosen  orbits — which 
here  also  are  determined  by  (99)  and  (100) — are  no  longer 
solely  dependent  on  the  sum  of  the  quantum  numbers  as  in 
the  case  of  the  non-relativistic  Kepler  motion,  but  that  the 
quantum  numbers  n  and  n  also  enter,  separately,  into  the 
expression  for  the  energy.  Only  as  a  first  approximation, 
therefore,  i.e.  when  the  relativity  correction  is  neglected, 
will  the  frequency  v  of  the  spectral  line  emitted  depend  on 
the  quantum  sums  s  +  s'  and  n  +  n'  alone,  as  (102)  shows. 
If  we  take  into  account  the  relativistic  change  of  mass  of  the 
electron,  on  the  other  hand,  v  will  also  depend  on  the 
individual  values  of  s,  s',  n,  w'.269  It  follows,  therefore,  that 
the  various  possibilities,  above  considered,  of  the  generation  of  a 
definite  spectral  line,  that  is,  the  passage  of  an  electron  from 
any  one  of  the  initial  orbits  s  +  s'  =  constant  to  any  one  of  the 
final  orbits  n  +  n'  =  constant,  no  longer  produce  exactly  the 
same  line,  but  give  rise  to  slightly  different  lines,  which,  how- 
ever, on  account  of  the  smallness  of  the  relativity  effect,  lie 
very  close  together.  This  is  Sommerfeld' s  explanation  of  the 
fine-structure  of  the  spectral  lines  in  the  case  of  the  hydrogen 
type  of  spectra.  For  example,  according  to  Sommerfeld,  the 
first  line  of  the  Balmer  series  (the  red  hydrogen  line  Ha)  must 
consist  of  five  components,  which  are  arranged  in  two  chief 
groups  (of  two  and  three  each).  The  mean  distance  of  these 
two  groups  from  one  another  should  amount,  according  to  the 
theory,270  to  about  0'126A ;  the  best  measurements  of  the 
hydrogen  doublet  gave  the  value  0'124A  (Paschen,  Mciasner). 


HIGHER  ELEMENTS  97 

If  this  agreement  already  speaks  strongly  in  favour  of  Sommer- 
f eld's  Theory,  the  exact  measurements,  by  F.  Paschen,™  of  the 
fine-structure  of  the  lines  of  positive  helium  (Fowler  series)  have 
given  a  still  more  convincing  proof  of  its  correctness ;  almost 
without  an  exception,  all  the  components  required  by  the 
theory  of  the  fine-structure  appeared  on  the  photographic  plate, 
and  thus  proved  strikingly  the  existence  of  the  stationary  paths 
of  the  electron  and  its  relativistic  change  of  mass. 

Two  interesting  consequences  may  yet  be  mentioned  here  ; 
they  are  directly  connected  with  Sommerfeld's  Theory  and 
Paschen's  observations.  First  of  all  they  have  rendered 
possible  the  use  of  the  fine-structure  measurements  for  a 
direct  "  spectroscopic "  determination  of  the  three  funda- 
mental constants  e,  m0  (mass  of  the  electron  at  infinitely  low 
speeds),  and  h.^2  Secondly,  K.  Glitscher™  was  able  to 
show  that  we  only  find  the  spectroscopic  observations,  for 
example,  the  size  of  the  hydrogen  doublet,  in  agreement  with 
the  theory,  when  we  use  for  the  variation  in  the  mass  of  the 
electron  the  formula  given  by  the  theory  of  relativity.  On 
the  other  hand,  Abraham's  Theory  of  the  rigid  electron  leads 
to  formulae  which  do  not  agree  with  experiment. 

§7.  Higher  Elements 

We  thus  see  that  Rutherford's  atomic  model  as  further 
developed  by  Bohr  and  Sommerfield  far  exceeded  the  ex- 
pectations which  it  could  reasonably  be  expected  to  fulfil.  At 
any  rate,  it  has  revealed  to  us  the  optical  series  of  hydrogen 
and  helium  with  undreamed-of  precision  as  far  as  the  finest 
details.  But  beyond  these  primary  gains,  it  has  undertaken 
a  further  series  of  successful  attacks.  Thus  Landew*  was 
successful  in  calculating  the  two  series-systems  of  neutral 
helium  (helium  and  parhelium)  by  taking,  in  contra- 
distinction to  Bohr,  a  model  of  the  neutral  helium  atom  in 
which  the  two  electrons  circle  around  the  double  positive 
nucleus  in  two  different  orbits,  either  co-planar  or  else 
inclined  at  an  angle  to  one  another.  In  this  case  then,  the 
external  electron,  the  leaps  of  which  generate  the  radiation, 
moves  in  a  field  in  which  the  simple  Coulomb  Law  no  longer 
holds,  on  account  of  the  disturbing  influence  of  the  inner 
electron.  Examples  of  this  type  which  differ  from  that  of 
7 


98  THE  QUANTUM  THEORY 

hydrogen  have  been  generally  investigated  by  Sommp.rfeld, 
who  has  shown 273  that  by  giving  up  the  Coulomb  field  we 
arrive,  to  a  first  and  second  approximation,  at  the  Bydberg 
and  Eitz  forms  of  the  series  laws.  A  very  promising 
beginning  in  setting  up  a  quantum  theory  of  the  spectral 
lines  was  thus  made. 


§8.  The  Stark  Effect  and  the  Zeeman  Effect  in  Bohr's  Theory 
of  the  Atom 

Under  the  circumstances  the  question  forces  itself  upon  us, 
whether  the  atomic  model  in  its  present  state  of  development 
is  able  to  account  for  the  Stark  effect,  that  is,  the  splitting  up  of 
the  spectral  lines  as  a  result  of  the  action  of  an  external  electric 
field  on  the  electrons  emitting  the  lines.  For,  as  we  may 
remember,  the  original  TJwmson  model  had  completely  failed 
just  at  this  point.  And  how  do  matters  stand  as  regards  the 
Zeeman  effect,  the  splitting  up  of  spectral  lines  as  a  result  of 
an  external  magnetic  field?  Could  the  new  model  explain 
these  phenomena  as  well  as  the  old  ?  Both  questions  have 
fortunately  been  answered  in  the  affirmative.  As  regards  the 
Stark  effect,  P.  S.  Epstein,™  in  an  important  paper,  succeeded 
in  demonstrating  the  following :  if  we  calculate  the  motion  of 
the  electron  under  the  influence  of  the  nucleus  and  the 
external  field,  according  to  the  methods  usual  in  celestial 
mechanics,  and  then  choose  from  among  all  mechanically 
possible  motions  the  allowable  stationary  orbits  by  applying 
the  modern  quantum  rules  for  several  degrees  of  freedom,  and 
if,  thirdly,  we  allow  the  electron  to  leap  from  one  of  these 
stationary  paths  into  another  (whereby  we  limit  the  infinite 
number  of  possible  passages  by  a  "principle  of  selection" 
presently  to  be  discussed),  then  the  Bohr  frequency  formula '(92) 
gives  with  the  most  admirable  accuracy  and  completeness,  both 
as  regards  position  and  number,  all  the  components  of  the 
resolved  lines  as  observed  by  Stark  in  the  cases  of  hydrogen 
and  positive  helium.  This  astonishing  result  must  be  re- 
garded as  a  further  strong  support  of  the  correctness  of 
Bohr's  model  and  its  system  of  quanta.  The  theory  of  the 
explanation  of  the  Zeeman  effect  has  up  to  the  present  not 
been  quite  so  successful,  It  is  true  that  Debyc™  and 


SELECTION  OF  RUBINOWICZ  AND  BOHR     99 

Sommerfield 278  have  been  able  to  derive  the  normal  Zeeman 
effect  (division  of  the  original  line  into  a  triplet  when  the 
line  of  observation  is  perpendicular  to  the  lines  of  force)  by 
calculation  from  the  model.  The  explanation,  however,  of 
two  important  phenomena  in  this  field  has  not  yet  been 
accomplished :  firstly,  the  anomalous  Zeeman  effect  and  its 
laws  (Runge-Preston  rule),  and  secondly,  the  fact,  discovered 
by  Paschen  and  Back,1219  that  even  in  the  case  of  lines  with  a 
complicated  fine-structure,  the  normal  triplet  is  formed  as 
the  magnetic  field  grows.  Further  investigation  will,  it  may 
be  hoped,  unravel  those  difficulties. 

§  9.  The  Principles  of  Selection  of  Rubinowicz  and  Bohr 

Inasmuch  as  the  foregoing  considerations  deal  only  with 
the  position  of  lines  in  the  spectrum,  i.e.  with  their  frequency, 
we  are  still  confronted  with  the  problem  of  their  form  of 
vibration,  i.e.  their  intensity  and  polarisation.  Moreover,  the 
important  question  had  yet  to  be  answered,  whether  all  leaps 
of  the  electron  from  any  one  stationary  path  to  any  other 
are  possible,  or  whether  the  number  of  allowable  transitions 
must  be  limited  by  some  "  principle  of  selection."  This 
also  is,  fundamentally,  a  question  of  intensity,  for  the  position 
may  be  regarded  as  follows  :  the  forbidden  transitions  corre- 
spond to  zero  intensity.  The  solution  of  this  whole  complex 
of  problems  has  been  greatly  advanced  quite  recently.  In 
the  first  place,  A.  Rubinowicz, &°  by  applying  the  law  of  the 
conservation  of  the  moment  of  momentum  (impuls-moment) 
to  the  system  atom  +  radiated  wave,  arrived  at  a  principle 
of  selection  and  a  rule  of  polarisation  of  the  following  form  : 
in  atoms  of  the  hydrogen  type,  which  are  removed  from  the 
influence  of  external  fields  of  force,  the  azimuthal  quantum 
number  n  of  the  electron  [see  formula  (99)]  can  only  alter  by 
0,  +1,  or  -  1,  when  emission  takes  place.  In  the  first  case, 
the  light  radiated  is  linearly  polarised,  in  the  two  other 
cases  circularly.  The  position  of  the  plane  of  the  orbit 
remains  unchanged  during  the  process  of  emission.  In 
the  case  of  atoms  differing  from  the  hydrogen  type,  and 
of  more  complicated  structure,  the  position  is  less  simple; 
if  we  set  the  total  moment  of  momentum  of  all  the 
masses  forming  part  of  the  system  (we  know  that  this 


100  THE  QUANTUM  THEORY 

impulse  remains  constant  during  the  motion),  equal  to  a 
whole  number,  n*,  times  ^,  it  is  just  the  changes  in  this 

number  n*  during  the  emission  which  must  be  limited  by  the 
principle  of  selection  in  the  same  manner,  as,  in  the  case 
above,  the  alterations  in  the  azimuthal  quantum  number  of 
the  individual  electron  in  its  leaps  were  limited.  Here  also, 
zero  change  in  the  azimuthal  quantum  number  gives  linear 
polarisation,  changes  by  +  1,  on  the  other  hand,  lead  to 
circular  polarisation.  In  place  of  the  orbital  plane  we  get 
the  "  invariable  plane  "  (at  right  angles  to  the  total  moments 
of  momentum  or  impulse-moments),  the  position  of  which  in 
space  remains  unaltered.  If,  finally,  the  atom  is  exposed  to 
an  external  field,  say  a  homogeneous  electric  field  (Stark 
effect)  or  a  homogeneous  magnetic  field  (Zeeman  effect),  then, 
as  we  know,  only  that  component  of  the  total  turning 
impulse  remains  constant  during  the  motion  of  the  masses 
forming  parts  of  the  atom  which  is  parallel  to  the  external 

field.     If  we  set  these  components  of  impulse  =  n^,  then 

only  the  alteration  of  this  number  n±  will  be  limited  by  the 
principle  of  selection  (that  is,  the  alterations  must  be  Ql  ±  1). 
The  principle  of  selection  is  thus  clearly  weakened  in  its 
action  by  the  external  field,  and  can,  if  fields  of  irregular 
strength  and  direction  act  on  the  atom,  become  completely 
illusory,  as,  for  example,  in  the  case  of  electric  discharges. 

By  means  of  entirely  different  considerations,  N.  Bohr  281 
arrived  at  results  which  coincide,  in  essentials,  with  those  of 
Eubinowicz,  but  exceed  them  greatly  in  range.  Bohr  started 
from  the  fact  that  in  the  limit  for  large  quantum  numbers, 
when  the  successive  stationary  states  of  the  atom  differ  very 
little  in  the  energy  they  involve,  the  frequency  that  the 
electron  emits  in  its  passage  between  neighbouring  states 
becomes  identical  with  the  rate  of  revolution  in  the  stationary 
orbit.282  The  electron  therefore  emits,  according  to  Bohr's 
frequency  condition,  the  same  line  that  it  sends  out  accord- 
ing to  the  classical  theory  of  electrons.  In  other  words,  for 
very  high  quantum  numbers,  the  quantum  theory  passes  over 
into  the  classical  theory.  (Bohr's  "Principle  of  Correspon- 
dence or  Analogy.")  Arguing  from  this  principle,  Bohr  pro- 


SELECTION  OF  RUBINOWICZ  AND  BOHR     101 

ceeds  as  follows  :  according  to  classical  mechanics,  the  motion 
of  the  electron  in  Bohr's  atom  may  be  represented  as  the  super- 
position of  component  harmonic  vibrations  of  the  frequency : 

"kl   =   Tl<»l    +   T20>2   +      .     .     .      +    TjWf       .  .         (103) 

Here,  TI  .  .  .  T/  are  whole  numbers  which  in  general  may 
have  all  values  between  —  oo  and  +  oo ;  the  o^  .  .  .  ay  are 
certain  constants  which  depend  on  the  character  of  the 
motion :  /  is  the  number  of  degrees  of  freedom.  Let  the 
amplitude  of  the  partial  vibration  characterised  by  the 
numbers  TJ  to  T/  be  ATi  .  .  .  Arf.  Then,  according  to  classical 
electrodynamics,  vki  is  the  frequency  of  the  radiated  partial 
wave  (T!  .  .  .  T,)  and  A^  .  .  .  A*f  is  a  measure  of  its  in- 
tensity. On  the  other  hand,  the  following  result  is  derived 
from  the  quantum  theory  (Bohr's  frequency  formula)  for  high 
quantum  numbers :  in  the  transition  from  an  initial  state 
characterised  by  the  quantum  numbers  mv  m2  .  .  .  w/  into  a 
final  state  corresponding  to  the  quantum  numbers  n^  .  .  .  Hf, 
a  line  of  frequency 

VQU  =  (%  -  n^  +  (ra2  -  w2)w2  +  .  .  .  +  (mf-  nf)u>f  .  .  .     (104) 

is  emitted.  Here  the  quantities  <o1  .  .  .  <D/  are  the  same 
constants  as  in  (103).  But,  according  to  Bohrs  Principle  of 
Analogy,  for  high  quantum  numbers  vki  =  VQU.  Hence  there 
follows  from  a  comparison  of  (103)  with  (104) 

Ti  =  mi  ~  ni>  T2  =  m2  ~  n2  •  •  •>  .  .  .  T/  =  ra/  -  w/  .  .  .  (105) 
i.e.  the  "  classical "  partial  vibration  (r{  .  .  .  T/)  corresponds  to 
that  quantum  transition,  in  which  the  quantum  numbers  alter 
by  exactly  TJ  .  .  .  T/.  The  polarisation  and  intensity  of  the 
wave  emitted  during  this  q^lantum  transition  may  be  calculated 
from  the  form  of  vibration  and  amplitude  of  the  "  corresponding 
classical"  partial  oscillation.  This  principle  which  has  been 
derived  for  high  quantum  numbers  is  extrapolated  by  Bohr 
with  great  boldness  over  the  region  of  all  quantum  numbers. 
Thus  the  important "  principle  of  correspondence  "  is  obtained. 
If  in  the  development  of  the  electronic  motion  in  terms  of 
partial  vibrations  the  term  (rlt  r2  .  .  .  ?/)  is  missing,  then 
the  corresponding  transition 


102  THE  QUANTUM  THEORY 

is  not  present.  Hence  there  follows,  for  example,  for  atoms 
of  the  hydrogen  type  in  a  field  free  from  force,  the  law  that 
the  azimuthal  quantum  number  can  in  all  emissions  only  change 
by  +  1  or  —  1,  both  of  which  lead  to  circularly  polarised 
radiation.  This  law  is  somewhat  more  limited  in  form  than 
that  of  Rubinowicz. 

Both  the  principles  of  selection  and  the  rules  for  the 
polarisation  and  the  intensity  have  stood  the  test  of  compari- 
son with  experiment.  Bubinowicz  himself  showed  that  his 
principle  of  selection  and  the  rule  of  polarisation  are  in  agree- 
ment with  Paschen's  measurements  of  the  fine-structure  of 
the  helium  lines,  and  further  with  the  observations  of  the 
Stark  effect  and  the  normal  Zeeman  effect.  P.  S.  Epstein  283 
and  H.  A.  Kramers™*  went  still  further,  and  were  able  to 
prove  by  profound  investigations,  based  on  Bohr's  Theory,  that 
the  calculations  of  intensity  along  the  lines  sketched  above 
were  also  in  surprising  agreement  with  observation.  Finally, 
Sommerfeld  and  Kossel 283  in  an  interesting  study  have  applied 
the  Rubinowicz  principle  of  selection  to  spectra  differing  from 
the  hydrogen  type  as  well,  and  have  shown  that  it  is  able  to 
explain  why  certain  series  appear  more  readily  and  are  more 
favoured  than  others,  as  it  were,  and  that,  by  the  selection  of 
the  "  possible  "  transitions,  it  sets  a  limit  to  the  multiplicity 
of  possible  combinations  in  a  manner  which,  so  it  appears, 
entirely  agrees  with  experience. 

§  10.  Collision  of  Electrons  on  the  Basis  of  the  Bohr  Atom 

While  in  this  way,  through  the  interpretation  and  unravell- 
ing of  the  universe  and  the  almost  bewildering  abundance  of 
spectroscopic  observations,  the  conviction  of  the  correctness 
of  Bohr's  atomic  model  deepened  more  and  more,  a  series  of 
observations  of  quite  another  kind  became  known  and  contri- 
buted considerably  to  the  consolidation  of  Bohr's  Theory. 
These  were  the  investigations  already  mentioned  earlier  in 
connexion  with  the  light-quantum  hypothesis,  which  dealt 
with  the  collision  of  free  electrons  with  gas  molecules  and 
atoms.  These  researches  were  conducted  particularly  by 
J.  Franck  and  G.  Hertz  28fl  and,  in  succession,  by  a  considerable 
number  of  American  investigators  in  a  systematic  manner. 
The  manifold  results  of  these  interesting  researches  may  be 


COLLISION  OF  ELECTRONS 


103 


sketched  here  schematically  by  a  simple  example.  What 
have  we  to  expect  when  electrons  collide  with  a  Bohr  Atom  ? 
As  a  simple  type  of  Bohr  atom,  let  us  choose  a  model  in  which 
z  electrons  revolve  around  a  2-fold  positively  charged  nucleus 
in  stationary  quantum  paths.  The  nature  and  spatial  arrange- 
ment of  these  paths,  as  well  as  the  distribution  of  the  electrons 
among  the  individual  paths  will  be  left  open,  and  we  shall 


FIG.  10. 

make  only  the  simplifying  assumption  that  one  electron — the 
so-called  valency  electron — revolves  alone  in  the  outermost 
orbit  (1)  (see  Fig.  10).  Let  this  be  the  "  normal,"  unexcited 
state  of  the  atom.  The  hydrogen  atom  (z  =  1)  is,  as  we  know, 
constituted  in  this  way,  and,  of  the  neutral  complicated  atoms, 
the  atoms  of  the  vapours  of  the  alkali  metals  (Li,  Na,  K,  Kb, 
Cs)  very  probably  also  fall  under  this  scheme.  If  by  any 
addition  of  energy  the  electron  is  "  raised  "  from  its  normal 


104  THE  QUANTUM  THEORY 

orbit  (1)  to  a  higher  orbit  (that  is,  one  having  more  energy), 
say  into  the  orbit  (2),  (3),  (4)  and  so  forth,  and  if  it  "falls" 
from  these  back  into  orbit  (1),  then  the  1,  2,  3  .  .  .  line  of  the 
so-called  "  Absorption-series  of  the  unexcited  atom  "  (principal 
series)  is  emitted.  The  frequencies  of  the  lines  emitted  are 
regulated  by  Bohr's  frequency  condition  (92),  i.e.  that  the  loss 
of  energy  Wn  —  Wl  incurred  in  passing  from  the  nib.  to  the 
first  orbit  is  equal  to  a  quantum  hvn>l  of  the  line  emitted : 

Wn-W^hv^.       .  .  .      (106) 

The  additional  energy  required  to  "  raise  "  the  electron  to  the 
higher  energy  level  can  be  obtained  in  two  ways :  firstly  by 
absorption  of  external  radiation ;  secondly  (and  that  is  the  case 
we  are  dealing  with  here)  by  electronic  impact.  If  external 
radiation  of  frequency  vn<l  falls  upon  the  atom,  a  quantum 
hvn,i  of  this  radiation  is  absorbed  and  is  used  to  raise  the 
electron  from  the  energy  level  Wl  to  the  higher  level  Wn 
=  Wl  +  hvn,i-  ^n  foiling  from  this  to  the  original  level,  the 
electron  then  emits  the  light  corresponding  to  the  line  absorbed. 
The  circumstance  is  further  noteworthy,  that  the  electron, 
when  it  is  raised  to  the  level  (2),  has  no  other  choice  than  to 
return  to  the  initial  level,  whereas  from  orbit  (4)  it  can  make 
one  of  three  possible  transitions — to  (3),  (2),  and  (1).  If, 
therefore,  the  atom  has  absorbed  light  corresponding  to  the 
line  j/2a  from  the  external  radiation,  it  will  re-emit  this  line 
with  its  full  complement  of  energy.  The  first  line  of  the 
absorption  series  is,  therefore,  in  contrast  with  all  other  lines, 
a  so-called  resonance  line. 

If  the  energy  required  -to  raise  the  electron  is  furnished  by 
tfce  impact  of  an  outside  electron,  then — as  Franck  and  Hertz 
were  the  first  to  prove — the  intruding  foreign  electron  will  be 
reflected  from  the  atom  perfectly  elastically  (according  to  the 
mechanical  laws  of  elastic  impact),  as  long  as  its  energy 
remains  below  a  certain  critical  value  ER.  If  this  energy 
value  is  reached,  the  impinging  electron  loses  all  its  energy, 
and  gives  it  up  to  the  electron  of  the  atom  which  has  been 
struck  ("  inelastic  impact ").  What  does  this  mean  according 
to  Bohr's  view  of  the  atom  ?  Obviously  ER  is  nothing  other 
than  W2  -  Wv  that  is,  the  energy  which  is  necessary  to  raise 
the  electron  from  its  normal  state  in  the  atom  to  the  orbit  (2). 


COLLISION  OF  ELECTRONS  105 

The  result  of  this  electronic  impact,  which  adds  energy  of 
amount  ER  to  the  atom  must  therefore  be  the  emission  of 
the  resonance  line.  If  this  view  represents  the  kernel  of  the 
matter,  then  the  energy  ER  must  be  connected  with  the 
frequency  i/2n  of  the  resonance  line  by  the  quantum  relation 

ER  =  hv^   ....     (107) 

This  relationship  has  been  excellently  verified  by  experiment. 
Thus  Tate  and  Footed  for  example,  find  in  the  case  of 
sodium,  that  the  first  inelastic  electronic  impact  takes  place 
when  the  impinging  electron  is  accelerated  by  a  potential 
of  VR  =  2-2  volts,  the  so-called  resonance  potential.  The 
energy  communicated  by  this  potential  to  the  impinging 
electron  is 


On  the  other  hand,  the  resonance  line  that  is  under  con- 
sideration here  is  the  D-line,  hence 


We  thus  see  that  the  relation  (107)  is  fulfilled  with  great 
accuracy.  The  same  holds  for  potassium  (Vp.  =  T55  volts, 
.-.  ER  =  2-47  -  10-12,  X21  =  7-685  •  10~5  /.  hvzl  =  2-55  -  1Q-12). 
In  the  case  of  the  inert  gases  (helium,  neon,  etc.)  and  the 
vapours  of  mercury,  zinc  and  cadmium,  similar  qualitative  and 
quantitative  relations  —  with  some  modifications  —  occur.  The 
excitation,  by  electronic  impact,  of  the  mercury  resonance  line 
A.  =  2-536  •  10  -5,  that  is  2-536.4,  discovered  by  Franck  and 
Hertz,  and  already  referred  to,  presents  a  charactei-istic 
example.  The  observed  resonance  potential  is  here  4-9  volts, 
while  from  the  relation 

v        300  p        300,  SQQhc 

VR  =  -  &R  =  -  AT.*  =  — 

e  e  eAj,! 

the  value  VR  =  4-86  volts  is  deduced. 

If  the  energy  of  the  impinging  electron  is  increased  beyond 
ER,  then  an  "inelastic"  impact,  accompanied  by  complete 
loss  of  the  energy,  is  to  be  expected  every  time  as  soon  as  E 


106  THE  QUANTUM  THEORY 

has  become  equal  to  Wn  -  Wl  (n  =  3,  4,  5  .  .  .).  By  these 
various  additions  of  energy  the  electron  attached  to  the  atom 
is  raised  successively  to  the  3rd,  4th,  5th  .  .  .  level  of  energy. 
If,  finally,  E  =  E&  =  W&  -  Wv  then  the  energy  of  the 
impinging  electron  is  just  sufficient  to  remove  the  electron 
attached  to  the  atom  to  infinity,  i.e.  to  ionise  the  atom.  E^ 
is  thus  the  ionisation  energy,  and  the  voltage  corresponding 

SOOT'1 
to  it,  FOQ  = 9P_  (  is  called  the  ionisation  potential.     From 

the  relation  (106)  we  get  immediately  the  important  equation 

E*  =  em  =/lv°0'1  •          •  (108) 

That  is  to  say,  the  ionisation  energy  is  equal  to  the  quantum 
which  corresponds  to  the  last  line  of  tJie  absorption  series,  that  is, 
to  the  "  series  limit."  This  quantum  relation  has  also  been 
excellently  confirmed  in  all  cases.  For  sodium,  for  example, 
Tate  and  Foote  found :  FQQ  =  5-13  volts,  which  gives  an 
ionisation  energy  of  the  value  E^  =  8'17  •  10~12.  On  the 
other  hand,  the  limit  of  the  principal  series  has  the  wave- 
length AQO  x  =  2-413  •  10- 5,  from  which  hv<x>  1  =  8-14  - 10'12, 
in  striking  agreement  with  the  value  of  E^  . 

For  mercury  vapour,  the  limit  in  question  of  the  principal 
series  A.^  =  1-188  •  10  - 5.  From  this  follows,  according  to 
(108),  FOQ  =  10-4  volts  while  the  measurements  of  various 
workers  gave  the  value  10*2  to  10'3  volts  (Tate,  Bergen, 
Davis  and  Goucher ;  Hughes  and  Dixon ;  Bishop™6).  From 
all  these  examples,  which  could  be  considerably  multiplied, 
the  conclusion  may  be  drawn  with  convincing  clearness  that 
the  Bohr  conceptions  have  laid  bare  the  nature  of  the  con- 
struction and  the  mode  of  action  of  the  atom  with  un- 
precedented lucidity. 

§11.   Einstein's  Deduction  of  Planck's  Law  of  Radiation  on  the 
Basis  of  the  Bohr  Atom 

Under  these  circumstances  the  suggestion  naturally  arises  to 
refound  the  law  of  black-body  radiation  by  taking  as  the  ele- 
mentary absorbing  and  emitting  structure  Bohr's  model  in 
place  of  the  linear  oscillator  used  by  Planck.  Einstein  289  has 
taken  this  step.  In  a  highly  important  study  he  investigated 


PLANCK'S  LAW  OF  RADIATION  107 

the  equilibrium  of  energy  and  momentum  between  black-body 
radiation  and  a  generalised  Bohr  model,  which,  stripped  of  all 
special  properties,  has  only  to  fulfil  the  quantum  condition  of 
being  able  to  assume  a  discrete  series  of  different  states.  For 
the  interaction  between  the  radiation  and  the  atom — absorption 
(Einstrahhing)  and  emission  (Ausstrahlung) — Einstein  intro- 
duces the  following  simple  hypotheses :  the  frequency  of  the 
emissions,  i.e.  the  transitions,  accompanied  by  loss  of  energy, 
of  the  atom  from  a  condition  (2)  of  higher  energy,  E%,  to 
a  condition  (1)  of  lower  energy,  Ev  shall  follow  the  same 
statistical  law  as  that  which  governs  the  disintegration  of 
radioactive  bodies,  i.e.  the  number  of  transitions  2  ->  1  in  the 
time  dt,  or,  as  we  may  say,  the  number  of  atoms  (2)  that  "  dis- 
integrate "  in  this  time  is  proportional  to  dt  •  Nv  where  N% 
denotes  the  number  of  atoms  momentarily  in  the  state  (2). 

But,  according  to  Einstein,  a  different  law  regulates  the 
processes  called  into  existence  by  the  effect  of  external  radi- 
ation. Under  the  influence  of  external  radiation  two  things 
may  happen :  either  an  atom  may  pass  from  state  (1)  to  state 
(2)  by  taking  up  energy,  this  is  the  "  proper  positive  absorp- 
tion." Or  the  case  may  also  occur,  that,  as  a  result  of  the 
phase-relation  between  the  field  of  the  external  radiation  and 
the  atom,  the  atom  loses  energy  through  the  action  of  the  im- 
pinging radiation,  and  hence  passes  from  state  (2)  to  state  (1) 
("negative  absorption").  The  rate  at  which  both  kinds  of 
transition  are  repeated  is  then  proportional  to  the  intensity 
Kv  of  the  external  radiation  :  the  number  of  transitions  1  ->  2 
associated  with  positive  absorption  in  the  time  dt  is  therefore 
proportional  to  N-^dtK.,, ;  the  number  of  transitions  2  -»  1  as- 
sociated with  negative  absorption  is  proportional  to  N2dtK.v. 
Here  N^  is  the  number  of  atoms  momentarily  in  the  state  (1). 
Nj_  and  N2  are  determined  by  the  laws  of  distribution  known 
from  the  theory  of  gases  and  statistical  mathematics  and  en- 
larged in  conformity  with  the  quantum  theory.  There  follows 
from  the  energy  equilibrium  between  in-coming  and  out-going 
radiation  at  the  temperature  T 

.        .        .     (109) 


108  THE  QUANTUM  THEORY 

where  k  is  Boltzmann's  constant,  and  A  is  a  constant  inde- 
pendent of  the  temperature.  From  Wieris  Displacement  Law 
(4)  it  follows,  firstly  that  A  is  proportional  to  v3  and  secondly 
that  E2  -  E!  is  proportional  to  v.  If,  therefore,  we  write 

E2  -  El  =  hv       . .        .        .    (110) 

we  recognise  in  this  expression  Bohr's  frequency  condition  (92). 
In  this  way  K,,  assumes  the  form  of  Planck's  Law  of  Eadia- 
tion,  arising  in  a  surprisingly  simple  and  elegant  manner  from 
a  minimum  of  hypotheses  of  a  general  character.  Einstein, 
in  pursuing  and  deepening  these  conceptions  by  writing  down 
the  expression  for  the  equilibrium  of  the  momenta  in  addition 
to  the  energies  of  the  in-coming  and  out-going  radiation,  was  led 
to  the  remarkable  conclusion  that  the  radiation  of  Bohr  atoms 
cannot  take  place  in  spherical  waves,  as  the  classical  theory 
of  electrons  requires,  but  that  the  process  of  emission  must 
have  a  particular  direction  like  the  shot  from  a  cannon.  We 
cannot  fail  to  recognise  that  this  brings  the  conception  that 
radiation  has  a  quantum-like  structure  (light-quantum  hypo- 
thesis) within  realisable  bounds. 


CHAPTEK  VII 
The  Quantum  Theory  of  Rontgen  Spectra 

§i.  The  Analysis  of  Rontgen  Spectra 

T)  AEALLEL  with  the  development  of  the  science  of  optical 
[  spectra,  a  theory  of  Eontgen  spectra  has  been  developed  of 
late  years  upon  the  same  basis.  This  theory  has  already  shed 
much  light  on  the  structure  of  atoms  and  thus  forms  a 
desirable  extension  of  the  theory  of  optical  spectra.  The 
investigations  of  Ch.  Barkla,  W.  H.  and  W.  L.  Bragg,  Moseley 
and  Darwin,  Siegbahn  and  Friman,'290  among  others,  have 
shown  that  by  the  impact  of  cathode  rays  upon  the  anti- 
cathode  of  a  Eontgen  tube  two  kinds  of  Eontgen  rays  arise : 
first,  the  so-called  "  impact  radiation  "  (Bremsstrahlung)  con- 
sisting of  an  extensive  and  continuous  range  of  wave-lengths 
(similar  to  the  continuous  background  of  visible  spectra) ; 
secondly,  the  "  characteristic  radiation,"  a  typical  line- spectrum, 
the  structure  of  which  depends  so  essentially  on  the  material 
of  the  anti-cathode  that  a  glance  at  this  spectrum  suffices  us 
to  deduce  immediately  and  unmistakably  the  nature  of  the 
material  of  which  the  anti-cathode  is  composed.  Thus  along- 
side the  optical  spectrum  analysis  of  Bunsen  and  Kirchhoff  a 
Eontgen-  or  X-ray  analysis  presents  itself.  It  has  further 
been  shown  that  the  characteristic  X-ray  spectrum  is  a 
purely  atomic  property,  and,  indeed,  an  additive  one.  If  we 
examine,  for  example,  the  X-ray  spectrum,  which  is  emitted 
by  an  anti-cathode  of  brass  (copper  +  zinc),  we  find  the 
lines  of  both  copper  and  zinc  unaltered  and  occupying  the 
same  positions  as  if  only  one  metal  were  present  in  turn.  No 
new  lines  appear.  Accordingly  we  are  led  to  suppose  that 
the  line-spectrum  arises  in  the  atoms  of  the  anti-cathode,  and 
is  generated  there  by  the  impinging  electrons  of  the  cathode 
109 


110  THE  QUANTUM  THEORY 

rays.  The  further  important  fact  appeared  that  the  lines  of 
the  characteristic  spectrum  may  be  arranged  in  series,  just 
like  those  of  the  optical  spectrum.  Thus  we  have  discovered 
up  to  the  present  a  short-wave  ^-series,  a  long-  wave  .L-series, 
and  a  still  longer-wave  M-series. 

The  most  curious  feature  of  these  spectra  is  their  connexion, 
by  a  definite  law,  with  the  atomic  number  of  their  element  in 
the  periodic  system.  If  we  plot  the  position  of  a  certain  line 
(say  the  first  line  Ka  of  the  ^-series)  for  the  successive 
elements  of  the  periodic  system,  a  perfectly  regular  progres- 
sive shift  is  revealed  :  the  line  advances  with  increasing 
atomic  number  steadily  towards  the  shorter  waves.  The  re- 
gularity of  this  advance  is  such  that  we  can  recognise  gaps  or 
false  positions  of  elements  in  the  periodic  system  immediately 
by  an  excessive  jump.  Now,  according  to  the  hypothesis, 
already  mentioned,  of  Eiitherford,  v.  d.  Broek,  and  Bohr,  the 
atomic  number  of  an  element  is  nothing  other  than  the 
number  of  its  nuclear  charge,  that  is,  the  number  of  elemen- 
tary positive  charges  of  its  nucleus.  If  to  this  we  add  the 
phenomenon  just  discussed,  according  to  which  the  steady 
advance  of  the  nuclear  charge  in  the  series  of  the  elements  is 
reflected  in  the  steady  displacement  of  the  X-ray  lines,  then 
we  are  forced  to  the  view  that  the  origin  of  tlie  X-ray  spectra 
must  be  localised  in  the  immediate  neighbourhood  of  the  nucleus, 
that  is,  in  the  inmost  part  of  the  atom.  For  in  this  region  the 
nucleus  clearly  has  the  greatest  power  and  is  least  disturbed 
by  external  electrons,  and  hence  it  is  here,  too,  that  the  growth 
of  the  nuclear  charge  will  make  itself  most  felt. 

The  connexion  between  the  position  of  the  X-ray  lines  and 
the  atomic  number  z  was  first  formulated  by  G.  Moseley.™ 
He  found  for  the  frequency  of  Ka  (first  line  of  the  JfT-series) 
and  La  (first  line  of  the  Z/-series)  the  empirical  relation 


(Ill) 


where  N  is  the  Eydberg  number. 

The  similarity  of  these  relations,  which  are  only  approxj- 


THE  ANALYSIS  OF  RONTGEN  SPECTRA     111 

mately  valid,  with  Bohr's  formula  (93)  for  the  series  of  the 
hydrogen  type  is  so  striking,  that  it  was  an  obvious  step  to 
seek  to  find  the  explanation  of  the  Eontgen  series  by  arguing 
on  the  basis  of  Bohr's  model. 

This  problem  was  attacked  chiefly  by  W,  Kossel,™2  A. 
Sommerfeld,™  L.  Vegard,™  P.  Debye,™  J.  Kroo,&*  and  A. 
Smekal.™1  And  thus,  in  addition  to  the  theory  of  the  optical 
spectra  which  take  their  origin  at  the  periphery  of  the  atom, 
a  theory  of  the  Kontgen  spectra  has  arisen  which  leads  us 


FIG.  11. 

into  the  inmost  regions  of  the  atom.  According  to  this  theory 
we  may  picture  to  ourselves,  in  general  terms,  the  emission 
of  the  Rontgen  spectra  as  follows  :  we  consider  a  neutral 
Bohr  atom,  consisting  of  a  2-fold  nucleus,  around  which 
z  electrons  revolve.  These  z  electrons  may  be  arranged  in 
different  rings.  The  innermost,  single-quantum  ring,  the  so- 
called  -ST-ring,  carries,  let  us  say,  pl  electrons  in  its  normal 
state ;  let  the  second  ring,  the  ir-ring,  be  a  two-quantum  ring 
occupied  by  p.2  electrons,  the  third,  three-quantum,  the  .M-ring 
with^3  electrons,  and  so  on  (Fig.  11).  The  question  whether 


112  THE  QUANTUM  THEORY 

we  can  reach  our  goal  with  this  conception  of  the  ring  by 
assuming  the  quantum  numbers  to  increase  as  we  go  outwards, 
and  whether  we  are  to  take  the  rings  as  co-planar  or  inclined 
to  one  another  will  be  left  open.  The  preparation  for  the 
emission  of  the  -BT-series  consists  in  this,  that  by  the  addition 
of  energy — whether  by  absorption  of  external  radiation  or  by 
electronic  impact — an  electron  of  the  K-ring  is  removed  to  in- 
finity, that  is,  the  atom  is,  so  to  speak,  ionised  "  inside,"  i.e. 
in  the  .BT-ring.  If  the  energy  of  the  atom  before  this  inner 
ionisation  =  W0,  and  after  the  ionisation  =  WK,  then  the 
amount  WK  -  W0  of  energy  must  be  provided.  Hence  every 
radiation,  the  energy  quantum  of  which  satisfies  the  condition 
hv  ^  WK  -  W0,  can  on  being  absorbed  effect  the  tearing  of  the 
electron  out  of  the  K-r'mg.  If  we  allow  the  v  of  the  external 
radiation  to  grow  slowly  from  small  values,  then,  at  the  point 

"TT7"       \JU 

VK  =  — *_ 9,  a  sudden  increase  of  the  absorption  occurs, 

because  from  this  point  onwards  the  external  radiant  energy 
is  used  for  the  "  ionisation  of  the  K-ring."  Thus  an  absorp- 
tion-band extends  from  v  =  VK  towards  higher  frequencies,  the 
edge  of  the  band  lying  at  VK.  This  phenomenon  of  the  "  edge 
of  the  absorption-band  "  has  already  been  interpreted  above 
in  the  sense  of  the  hypothesis  of  light-quanta.  If  the  addition 
of  energy  is  provided  by  the  impact  of  a  strange  electron, 
coming  from  without,  then  its  energy  must  be  E>  WK  -  W0, 
that  is,  E  ^>  ~hvK,  a  relation,  which  we  have  already  deduced 
earlier  from  the  standpoint  of  the  quantum  hypothesis  of  light. 
By  ionisation  of  the  .ST-ring  the  atom  is  now  prepared  for 
^-emission.  If  now  an  electron  falls  from  the  2-quantum 
Zv-ring  into  the  1-quantum  J5"-ring,  filling  up,  so  to  speak,  the 
gap  produced  there,  then  the  first  line  of  the  K-  series,  Ka, 
will  be  emitted.  If  on  the  other  hand  the  gap  in  the  .ST-ring 
is  filled  by  an  electron  of  the  3 -quantum  If -ring,  or  the 
4-quantum  .W-ring,  Kp  or  Ky  result  respectively.  The  position 
is  quite  analogous  as  regards  the  L-  and  .M-series.  If,  by  the 
addition  of  energy  (absorption  or  electron-impact),  an  electron 
of  the  Zi-ring  is  battered  off,  that  is  if  the  L-ring  is  ionised, 
then  the  atom  is  prepared  for  the  emission  of  the  ZJ-series. 
If,  now,  the  gap  in  the  2-quantum  L-ring  is  filled  by  an 
electron  of  the  3-quantum  M-ring,  the  first  line  of  the  Z/-series, 


FINE-STRUCTURE  OF  RONTGEN  LINES     118 

La,  results;  if  it  is  filled  by  an  electron  of  the  N-riug,  the 
second  line  of  the  .L-series,  Ly,  results  (the  notation  is  not 
quite  consistent  but  will  serve  the  present  purpose),  and  so 
forth. 

The  converse  phenomenon  to  line  emission,  viz.  line  absorp- 
tion, with  which  we  are  acquainted  in  visible  spectra,  appears 
at  first  sight  to  be  missing  here.  That  is,  however,  as  W. 
Kossel 298  recently  showed,  an  error.  It  is  true  that  the  ejected 
electron  of  the  .ST-ring,  for  example,  cannot  in  general  be 
caught  upon  the  L-,  M-,  or  .N-ring,  because  all  places  on  them 
are  already  occupied.  An  absorption  of  the  lines  Ka,  Kp,  Ky, 
is  therefore  in  this  case  impossible.  But  the  electron  of  the 
K-ring  can  certainly  come  to  rest  on  an  unoccupied  quantum 
orbit  outside  the  occupied  rings,  that  is,  outside  the  surface 
of  the  atom.  In  this  process  a  "  line  "  is  actually  absorbed, 
namely,  that  line  of  which  the  hv  is  equal  to  the  energy- 
difference  between  the  K-rmg  and  the  final  orbit  of  the  ejected 
electron.  This  refinement  of  our  considerations  shows,  then, 
that  the  electron  from  the  .fiT-ring  does  not  need  to  be  raised 
immediately  to  infinity,  but  that  line  absorptions  may  occur 
before  the  edge  of  the  band  of  absorption  is  reached. 

§2.  The  Fine-structure  of  Rontgen  Lines 

It  is  particularly  noteworthy  that  Sommer/eld  succeeded 
also  in  the  field  of  X-ray  spectra  in  explaining  the  fine- 
structure  of  the  lines  by  calling  in  the  aid  of  the  theory  of 
relativity.  Thus,  for  example,  the  2-quantum  .L-orbit  is 
"double";  it  can  occur  as  a  circle  (n  =  0,  n  =  2)  or  as  an 
ellipse  2"  (n  =  1,  n  =  1).  Hence  the  line  which  is  emitted 
by  the  electron  of  which  the  L-rmg  is  the  initial  orbit,  namely, 
Ka,  is  a  doublet  (Ka  and  Ka-).  In  just  the  same  way,  those 
lines  for  which  the  Iv-orbit  is  the  final  orbit  of  the  electron 
are  doublets,  namely,  the  line  La  (more  exactly  La>)  to  which 
Lp  is  added  to  make  a  doublet ;  further,  Ly  which  forms 
a  doublet  with  L&,  and  so  forth.  The  distance  between  the 
components  of  the  doublets  (expressed  in  frequencies)  comes 
out,  according  to  Sommerfeld's  Theory,  as  approximately  pro- 
portional to  the  fourth  power  of  the  atomic  number  z.  Hence 
here,  in  the  X-ray  region,  where  we  are  dealing  for  the  most 
part  with  elements  having  fairly  high  atomic  numbers,  the 
8 


114  THE  QUANTUM  THEORY 

doublets  appear  microscopically  enlarged  as  compared  with 
the  microscopic  hydrogen-doublet  (z  =  1).  During  the  emis- 
sion of  X-rays  the  electron  approaches  very  near  to  the 
highly-charged  nucleus,  and  hence  the  relativistic  effects  of 
the  resolution  of  the  lines  are  much  greater  than  in  the  case 
of  the  optical  spectra,  in  which  the  electron  is  moving  at  the 
surface  of  the  atom,  where  it  is  almost  entirely  screened  from 
the  action  of  the  strong  nucleus  by  the  remaining  electrons. 
With  the  help  of  the  following  relation  deduced  theoretically 
and  adapted  to  experimental  evidence, 

<•  -  fr")'     •     -.<112> 


Sommerfeld  was  able  to  calculate  the  hydrogen-doublet  from 
the  observed  L-doublets,  and  compare  it  with  the  results  of 
experiment.  The  agreement  is  very  satisfactory. 

§  3.  The  Distribution  of  Electrons  among  the  Rings.     Objections 
to  the  Ring-arrangement  of  Electrons 

The  quantitative  calculation  of  the  simplest  case,  namely, 
the  emission  of  Ka,  led  Debye  to  the  conclusion  that  the 
.ST-ring  in  the  normal  state  consists  of  three  electrons.  To 
this  Kroo,  by  elaborating  the  calculation,  adds  the  con- 
clusion that  the  L-r'mg  contains  in  its  normal  state  nine 
electrons.  With  these  two  distribution  numbers,  pl  =  3, 
p2  =  9,  the  position  of  Ka  could  be  represented  as  a  function 
of  the  atomic  number  z  for  all  elements.  The  emission  of  Ka 
takes  place  according  to  the  following  obvious  scheme  : 


I  K-ring     L-ring 


Normal  state  |      3  9    .  _     .      .        ,  ,      _   . 

Initial  state    [     2     |     9    >  I°ni8atlon  of  «*  Z-nng. 
KEafBtate      |      8     |     8    > EmiSS1On  °f  *" 

The  two  distribution  numbers  (Besetzungszahleri)  thus  found 
for  the  two  innermost  rings  excite  our  attention.  For  on 
the  basis  of  the  Periodic  System  with  its  periods  of  eight 
we  ought  to  expect,  according  to  Kossel,  the  numbers  2  and  8. 


DISTRIBUTION  OF  ELECTRONS  115 

The  strange  occurrence  of  the  numbers  3  and  9  becomes 
an  objection,  when  we  consider  the  case  of  sodium  (z  =  11). 
Here,  according  to  Kossel,  we  should  expect  the  numbers  2, 
8,  1,  since  in  all  probability  an  electron  (the  valency  electron) 
revolves  alone,  as  in  the  case  of  all  alkali  metals,  around  the 
outside  quantum  orbit  (M-ring).  In  any  case  it  is  impossible 
that  the  two  innermost  rings  together  should,  in  the  normal 
state,  contain  12  (=  3  +  9)  electrons.  If  we  attempt  to  go 
a  step  further  still  on  the  basis  of  Kroo's  numbers  3  and  9, 
and  to  set  up  a  formula  which  represents  for  all  2*8  the 
position  of  La  in  conformity  with  observation,  and  thereby 
to  determine  the  number  of  electrons  p3  on  the  .M-ring,  we 
find,  as  A.  Smekal300  showed,  that  this  mode  of  representation 
is  impossible  with  any  combination  3,  9,  py  Nor  do  we  fare 
better  if  we  incline  the  various  rings  to  one  another,  and  take 
their  interaction  into  account.  The  suspicion  is  forced  upon 
us,  that  perhaps  the  whole  conception  of  the  arrangement 
into  plane  rings  does  not  correspond  with  fact,  but  that,  rather, 
the  electrons  in  the  atom  form  spatially  symmetrical  figures. 
This  suspicion  is  very  much  strengthened  by  a  series  of  pro- 
found investigations  carried  out  by  M..  Born  and  A.  Landd.901 
Following  on  M.  Bom's  investigations  of  the  dynamics  of 
the  crystal-lattice,  which  we  discussed  in  detail  earlier  hi 
connection  with  the  atomic  heat  of  solids,  the  two  in- 
vestigators asked  themselves  the  question,  whether  it  is 
possible  to  build  up  the  cubic  crystal-lattice  of  the  alkaline 
halides  (NaCl,  NaBr,  Nal;  KC1,  KBr,  KI,  etc.)  from  ions  of 
Bohr  atoms,  by  taking  into  account  only  the  mutual  electro- 
static forces;  and  whether  this  method,  if  possible,  would 
enable  them  to  prophesy  the  crystal  properties  (lattice-con- 
stant, compressibility)  from  the  atomic  models  of  the  two 
constituent  ions.  The  answer  to  this  question  has  been,  on 
the  whole,  in  the  affirmative.  But  when  the  calculation  of 
the  compressibility  of  these  crystals  was  carried  out,  the 
remarkable  result  manifested  itself  that  crystals  are  found  to 
be  too  soft,  that  is,  insufficiently  rigid,  if  the  conception  of  the 
ring-arrangement  of  electrons  in  the  atom  is  maintained.  On 
the  other  hand,  we  get  good  agreement  with  the  observations 
if,  following  Born,  we  introduce  the  hypothesis  that  the 
electrons  are  arranged  spatially.  A  complex  of  eight  electrons, 


116  THE  QUANTUM  THEORY 

as  occurs  in  sodium,  potassium,  etc.,  does  not  therefore  occupy 
a  plane  8-ring ;  the  eight  electrons  describe  paths  of  ciibical 
symmetry.  Into  the  still  obscure  region  of  these  "  spatial " 
electron  paths,  A.  Lande302  has  made  some  successful  in- 
cursions. 

From  all  that  has  been  said  it  would  appear  to  be  certain 
that  in  dealing  with  Rontgen  spectra,  too,  we  can  no  longer  be 
content  with  the  arrangement  of  the  electron  rings  in  planes, 
and  that  the  whole  quantitative  theory  of  the  Eontgen  series, 
including  Sommerfeld's  fine-structure  of  the  K-  and  the  L- 
doublets,  must  be  built  up  on  a  fresh  foundation. 


CHAPTEB  VIII 

Phenomena  of  Molecular  Models 
§  i.  Dispersion  and  Magneto-rotation  of  the  H2  Molecule 

WHILE  the  X-ray  spectra  and  the  spectra  of  the  optical 
series  arise  from  the  atoms  of  the  elements  (and  hence 
their  theory  links  up  with  the  atomic  models),  there  is  a  series 
of  phenomena  which,  in  the  case  of  polyatomic  substances, 
are  peculiar  to  the  molecules,  and  the  theory  of  which, 
therefore,  is  founded  on  the  molecular  models.  Chief  among 
these  are  the  normal  dispersion,  the  rotation  of  the  plane  of 
polarisation  in  the  magnetic  field  (magneto-rotation),  and, 
further,  the  great  and  complicated  subject  of  band-spectra. 
Up  till  a  few  years  ago,  dispersion  and  magneto-rotation  had 
been  exclusively  treated  from  the  standpoint  of  the  Thomson 
model,  that  is,  with  the  help  of  quasi-elastically  bound 
electrons,  and  this  explanation  had  served  in  turn  as  a 
powerful  support  for  this  model.  Nevertheless,  discrepancies 
in  these  theories  had  long  been  known.  For  example, 
measurements  calculated  upon  the  basis  of  the  dispersion 
theories  of  Drude,  Voigi,  or  Planck  led  to  values  for  the  ratio 

of  the  charge  to  the  mass  of  the  electron  ( — )  which,  in  com- 
parison with  the  direct  measurements  of  this  quantity  (based 
upon  the  deflection  of  the  cathode-  or  /3-rays  in  the  electric 
and  magnetic  fields)  which  were  much  too  small.  When, 
however,  the  Thomson  model  became  displaced  by  the 
Rutherford-Bohr  model,  and  the  successes  of  the  Bohr  atomic 
model  increased  at  an  undreamed-of  rate,  the  question  arose 
whether  an  unobjectionable  theory  of  dispersion  and  magneto- 
rotation  could  not  be  founded  upon  these  new  views.  The 
difficult  position,  into  which  we  are  brought  by  this  problem, 
117 


118  THE  QUANTUM  THEORY 

arises  from  the  fact  that  we  do  not  actually  know  a  single 
instance  of  the  exact  manner  in  which  a  polyatomic  Bohr 
molecule  is  built  up  from  its  nuclei  and  electrons.  The 
exact  knowledge  of  this  structure,  and  the  motion  of  all  the 
electrons  is  absolutely  necessary,  if  we  desire  to  know  how 
the  molecule  reacts  upon  external  waves  (dispersion).  It  is 
true  that  W.  Kosseiw*  has,  in  a  detailed  study  already 
referred  to  above,  pointed  out  the  general  guiding  lines  along 
which,  from  the  chemical  point  of  view,  the  building-up  of 
the  atom  from  molecules  must  be  carried  out,  but  the  details 
of  this  construction  remain  open.  Only  in  a  few  of  the 
simplest  cases  have  detailed  molecular  pictures  been  con- 
structed and  closely  tested.  Thus  Bohr,  as  we  remarked  in 
discussing  the  atomic  heat  of  gases,  has  already  proposed  a 
model  of  the  diatomic  hydrogen  molecule.  It  has  the  follow- 
ing construction  (see  Fig.  8) :  two  singly-positive  nuclei  (that 
is,  each  consisting  of  only  a  single  positive  charge)  are 
separated  by  the  distance  26.  In  the  vertical  plane  which 
bisects  the  line  joining  the  nuclei,  two  electrons  rotate, 
diametrally  opposite  one  another,  on  a  circle  of  diameter  2a. 
The  equilibrium  of  the  Coulomb  and  the  centrifugal  forces 
requires  that  a  =  b^/3.  By  means  of  this  relation,  and  by  the 
quantum  condition  that  each  electron  must  have  the  moment 

of  momentum  — ,  the  model  is  completely  determined  in  all 

2ir 

its  dimensions  and  speeds.  It  was  this  model  which  was 
the  first  to  be  proposed :  it  was  examined  by  P.  Debye  3°* 
with  reference  to  its  dispersion.  On  account  of  its  sym- 
metrical structure  the  molecule  possesses  no  electrical  mo- 
ment in  its  normal  state.  If,  on  the  other  hand,  it  is  struck 
by  an  external  light  wave,  the  motion  of  its  electrons  is 
periodically  disturbed ;  they  depart  from  the  normal  quantum 
path,  fall  into  forced  vibration,  and  thus  generate  an  electric 
moment  which  changes  periodically  in  step  with  the  external 
wave.  Thus  the  original  motion  of  the  primary  wave  is 
changed,  and  dispersion  results.  We  may  conceive  this  as 
follows :  Let  c  be  the  velocity  of  the  primary  wave  in  vacuo. 
The  oscillations  of  the  electrons  generate  a  secondary  wave 
which  spreads  out  from  the  molecules.  All  these  secondary 
waves  combine  with  the  primary  wave  to  a  form  new  wave 


OBJECTIONS  TO  BOHR'S  MODEL          119 

which  moves  with  the  altered  velocity  q,  the  value  of  which 
depends  on  the  frequency  of  the  primary  wave.  But  just 
this  is  the  phenomenon  of  dispersion.  The  electronic  vibra- 
tions which  occur  here  are  not  oscillations  about  positions  of 
equilibrium,  as  in  the  case  of  the  quasi-elastic  model,  but 
oscillations  about  stationary  paths.  Moreover,  here,  the  force 
holding  the  electrons,  as  opposed  to  the  usual  classical 
theories  of  dispersion,  is  anisotropic  (that  is,  the  electron  is 
held  by  different  forces  in  different  directions) ;  above  all,  by 
means  of  this  anisotropy,  it  was  possible  to  explain  away  the 

disagreement   in   the   value  of  — ,  which  had  previously  been 

WIC 

found  to  be  too  small ;  and  Debye  succeeded,  on  the  basis  of 
the  normal  value  of  — ,  in  deducing  from  the  theory  the 

observed  dispersion  curve  of  hydrogen,  that  is,  the  curve 
which  shows  how  its  coefficient  of  refraction  depends  on  the 
wave-length.  It  should  be  noted  that  in  the  formula  for  the 
coefficient  of  refraction,  no  single  constant  is  arbitrary,  but 
that  the  dispersion  formula  is  made  up  entirely  of  universal 
constants. 

Using  the  same  method  (calculus  of  disturbances),  P. 
Scherrer30*  has  calculated  the  rotation  of  the  plane  of 
polarisation  which  linearly  polarised  light  undergoes  in  its 
passage  through  hydrogen  under  the  influence  of  a  magnetic 
field.  His  efforts  were  equally  successful. 

§  2.  Objections  to  Bohr's  Model  of  the  Hydrogen  Molecule 

In  spite  of  the  successes  which  the  Bohr  model  of  the 
hydrogen  molecule  has  won,  a  list  of  weighty  objections  to 
it  has  accumulated  in  the  course  of  time.  That  the  con- 
tribution which  the  rotation  (more  accurately,  the  regular 
precession)  of  this  molecule  makes  to  the  molecular  heat  at 
low  temperatures,  does  not  correspond  with  the  observations 
of  Eucken,  has  been  shown  by  P.  S.  Epstein,  as  we  have 
already  mentioned.  Also  at  high  temperatures,  when  the 
oscillations  of  the  two  nuclei  relatively  to  one  .another  con- 
tribute to  the  molecular  heat,  no  agreement  between  theory 
and  observation  has  been  found  in  the  case  of  the  Bohr  model, 
as  G.  Laski 306  recently  showed, 


120  THE  QUANTUM  THEORY 

Further,  the  model  must  possess,  in  consequence  of  the 
revolving  electrons,  an  almost  fixed  magnetic  moment  parallel 
to  the  axis  of  the  nucleus,  that  is  to  say,  it  must  be  equivalent 
to  a  molecular  elementary  magnet,  which  endeavours  to  set 
itself,  in  an  external  magnetic  field,  parallel  to  the  lines 
of  force.  Hydrogen  ought,  therefore,  to  be  paramagnetic, 
whereas  it  is  diamagnetic. 

Another  very  important  objection,  to  which  Nernst  in 
particular  drew  attention,  is  the  following :  if  we  calculate 
the  work  which  is  necessary  to  separate  the  molecule  into 
its  two  atoms,  the  so-called  heat  of  dissociation,  we  get  s07  the 
value,  61,000  calories.  On  the  other  hand,  Langmuir  308  found 
84,000  cals.,  Isnardi**  95,000  cals.,  /.  FrancJc,  P.  Knipping 
and  Thea  Kriiger™  81,000  (±  5700)  cals.  In  any  case,  the 
calculated  heat  of  dissociation  comes  out  25  per  cent,  too 
small.sioa 

Finally,  W.  Lenz 311  has  recently  increased  the  objections 
to  the  hydrogen  model  by  an  important  one  based  on  a 
theory  of  band-spectra,  which  we  shall  discuss  below.  He 
proved  that  the  band-lines  of  hydrogen  and  nitrogen  can 
exhibit  the  observed  Zeeman  effect,  only  if  these  molecules 
possess  no  moment  of  momentum  around  the  nuclear  axis. 
The  fact  that  the  two  electrons  in  Bohr's  molecular  model 
revolve  in  the  same  sense,  however,  endows  it  with  just  such 
a  moment  of  momentum.  On  the  whole,  the  Bohr  model  does 
not  seem  to  correspond  to  reality ;  the  arrangement  of  the  two 
nuclei  and  electrons  must  plainly  be  quite  different.  No 
satisfactory  model,  however,  has  yet  been  found. 

§3.  Models  of  Higher  Molecules 

Matters  are  no  better  in  the  case  of  models  of  the  more 
complicated  molecules.  It  is  true  that  Sommerfeld 312  and 
F.  Pawer313  have  also  worked  out  the  theories  of  dispersion 
and  magneto-rotation  in  the  case  of  the  more  general  Bohr 
models  (N2  and  O2)  which  are  constructed  on  the  lines  of  the 
hydrogen  model.  According  to  Sommerfeld,  four  electrons 
revolve  about  the  line  joining  the  two  nuclei  in  the  case  of 
oxygen,  each  of  which  acts  with  an  effective  charge  +  2e ;  in 
the  case  of  nitrogen,  a  ring  of  six  electrons  rotates  about  the 
nuclear  axis,  while  the  nuclei  carry  triple  effective  charges. 


QUANTUM  THEORY  OF  BAND-SPECTRA     121 

Sommerfeld  was  able  to  obtain  agreement  with  observation  only 
by  setting  up  for  each  electron  of  a  valency  ring  of  2s-electrons 
the  unaccountably  strange  quantum  condition :  moment  of 

momentum  =  ^  ^s>  undoubtedly     a     most     unsatisfactory 

result.  Gerda  Laski31*  obtained  better  results  with  some- 
what different  models,  which  she  chose  in  such  a  way  that 
the  specific  heat  of  the  two  gases  at  high  temperatures  agreed 
with  the  observations  of  Pier.318  According  to  her  ideas,  the 
nitrogen  molecule  must  consist  of  two  seven-fold  positive  nuclei, 
each  of  which  is  closely  surrounded  by  a  1-quantum  ring  of 
two  (or  three)  electrons.  The  "  valency  ring  "  in  the  central 
vertical  plane  is  2-quantum  and  contains  ten  (or  eight) 
electrons.  Analogously,  the  oxygen  molecule  consists  of  two 
eight-fold  positive  nuclei,  each  encircled  by  a  1-quantum  ring 
of  two  (or  three)  electrons,  whereas  the  2-quantum  valency 
ring  contains  twelve  (or  ten)  electrons.  The  same  objections 
apply  to  some  extent  to  these  models  of  Sommerfeld  and  Laski 
as  to  the  hydrogen  model.  For  example,  they  give  no  account 
of  why  oxygen  should  be  paramagnetic,  and  nitrogen,  on  the 
other  hand,  diamagnetic.  Moreover,  the  above-mentioned 
objection  of  Lenz  applies  in  full  force  to  these  models ;  for 
they  all  possess  moments  of  momentum  around  the  nuclear 
axis.  In  conclusion,  we  feel  bound  to  admit  that  the  exact 
constitution  of  even  the  simplest  models  is  at  present  unknown 
to  us. 

§  4.  The  Quantum  Theory  of  Band-spectra 

To  conclude  this  chapter,  we  shall  turn  our  attention  to  the 
band- spectra,  and  collect  together  shortly  what  the  quantum 
theory  has  been  able  to  assert  about  them  up  to  the  present 
time.  That  they  belong  to  molecules  and  compounds  may 
nowadays  be  regarded  as  certain.  The  first  attempt  to  con- 
struct a  logical  quantum  theory  of  band-spectra  was  under- 
taken by  K.  Schwarzschild 316  who  clearly  recognised  the 
importance  of  the  rotation  of  the  molecule  in  the  production 
of  these  spectra.  His  conceptions  may  be  defined  as  follows  : 
a  system  of  electrons  revolves  at  a  definite  quantum  distance 
around  a  molecule  which  itself  rotates  according  to  quantum 
conditions,  the  assumption  being  made  for  the  sake  of 


122  THE  QUANTUM  THEORY 

simplicity  that  the  motion  of  the  electrons  is  not  influenced 
by  the  motion  of  the  molecule.  If  E0  is  the  quantum  energy 
of  the  electrons,  Er  the  quantised  rotational  energy  of  the 
molecule,  then  E0  +  Er  =  E  is  the  total  energy  of  the  system. 
If  the  three  chief  moments  of  inertia  of  the  molecule  /  are 
equal  to  one  another,  then  it  follows,  just  as  in  (80),  that 


where  n  denotes  the  rotational  quantum  number.     Therefore 


If,  now,  the  system  passes  from  one  quantum  state  having 
the  electronic  energy  EQ  and  the  rotational  quantum  number 
n  into  another  quantum  state  having  the  electronic  energy  E'0 
and  the  rotational  quantum  number  n',  then  it  follows  from 
Bohr's  frequency  formula  (92)  that  the  frequency  of  the  line 
radiated  is  given  by 

_  E0-E'0      (n»  -  n> 

7  T  o     2  T  '  '        V     A    / 

If  we  keep  all  the  quantum  numbers  which  occur  here,  except- 
ing n,  constant,  and  allow  n  to  vary,  then  we  get  a  series  of 
lines  progressing  towards  the  violet  and  having  the  frequencies 

v  =  a  +  bn2         (a  and  b  are  constants)       .     (115) 

This  is  a  formula  which  had  already  been  given  empirically 
by  Deslandres,*11  and  which  is  approximately  true  for  the  lines 
of  many  bands. 

Following  Schwarzschild,  T.  Heurlinger  318  and  W.  Lenz*19 
in  particular,  have  further  developed  and  refined  the  quantum 
theory  of  band-spectra.  For  example,  Lenz  has  pictured  the 
molecule  as  a  symmetrical  top  having  two  moments  of  inertia 
and  a  rotational  rigidity  (moment  of  momentum)  around  the 
axis  of  the  figure,  and  hence  deals  from  the  outset  with  a  regular 
precession  of  the  molecule  in  place  of  a  rotation.  Using 
Bohr's  frequency  formula,  and  applying  the  principles  of 
selection,  he  obtained  the  following  general  foi-mula  for  the 
lines  of  a  band  : 

v  —  a  +  bn  +  en2        (a,  bt  c  are  constants)      .     (116) 


QUANTUM  THEORY  OF  BAND-SPECTRA     123 

which  is  obeyed,  according  to  Heurlinger,  in  the  case  of  the 
so-called  "  cyanogen  "  lines  of  nitrogen,  for  example.  In 
addition  to  the  lines  given  by  (116),  Lenz's  Theory  requires  the 
occurrence  of  the  series  given  by  the  formula 

+        +  ™2     .        .        .     (117) 


for  the  case  that  the  molecule  really  possesses  a  finite  moment 
of  momentum  about  its  axis  of  figure.  A  series  which  follows 
this  law  does  not,  however,  exist  in  the  cyanogen  bands,  ac- 
cording to  Heurlinger.  Lenz  deduces  from  this  the  conclusion 
already  mentioned,  that  the  nitrogen  model  does  not  possess 
a  rotational  rigidity  about  its  axis.  By  calculating  the 
Zeeman  effects  of  the  band  lines,  and  comparing  them  with 
observation,  Lenz  was  able  to  confirm  this,  and  to  extend  it  to 
the  hydrogen  molecule. 

The  infra-red  Bjerrum  absorption  bands  of  the  diatomic  and 
polyatomic  gas  compounds,  which  we  had  discussed  at  length 
in  Chapter  Y,  belong  to  the  general  type  of  band-spectra.  If 
we  are  to  deduce  them  from  a  theory  consistently  founded  on 
quanta  —  and  not,  as  we  did  earlier,  half  according  to  the 
quantum,  half  according  to  the  classical  theory  —  we  must 
follow  closely  the  course  pursued  above,  with  the  difference 
that,  in  place  of  the  energy  of  the  electronic  system  there  will 
appear  the  energy  of  the  atoms,920  with  which  the  rotational 
energy  of  the  molecule  is  combined,  as  a  first  approximation, 
additively.  The  logical  carrying  out  of  this  calculation  (in 
which  Bohr's  frequency  formula  and  the  principle  of  corre- 
spondence are  applied),  which  was  undertaken  by  Heurlinger  &l 
and  the  author,322  gives  for  the  structure  of  the  "  fluted  "  ab- 
sorption bands  an  arrangement  of  lines  which  at  first  sight 
does  not  appear  to  agree  with  the  beautiful  and  exact  measure- 
ments of  Imes.323  The  theory  gives  for  the  position  of  the 
absorption  lines  a  formula 

l/  =  Vo±(n  +  i)JL,        (n=l,2,3...)     (118) 

and  therefore  requires  that  all  neighbouring  lines  be  equi- 
distant, including  the  two  in  the  middle  (n  =  0).  On  the  other 
hand,  Imes'  observations  show  with  indubitable  clearness  that 
the  interval  between  the  two  middle  lines  is  twice  as  great  as 


124  THE  QUANTUM  THEORY 

the  interval  between  all  neighbouring  lines.  This  apparent 
contradiction  is  explained,  as  A.  Kratzer32*  recently  showed, 
in  a  surprising  fashion,  if  we  take  into  account  the  intensity 
of  the  absorption  lines  according  to  Bohr's  Principle  of  Analogy. 
For  it  then  appears  that  the  first  absorption  line  to  the  right 
of  the  middle  v0-line,  namely,  the  line 

h 

"  =  "o  +  SW 

(which  is  derived  from  formula  (118)  by  setting  n  =  0  and 
using  the  positive  sign  for  the  second  term)  is  of  vanishingly 
small  intensity.  This  line  is  generated  when  the  molecule 
passes  over  from  an  initial  rotationless  and  vibrationless  state 
into  the  final  state  in  which  the  two  ions  oscillate  relatively 
to  one  another  with  one  quantum,  and  in  which,  at  the  same 
time,  the  molecule  rotates  as  a  whole  with  one  quantum.  The 
rotationless  and  vibrationless  state  has,  however,  a  vanishingly 
small  probability  ;  the  number  of  transitions  from  this  initial 
state  per  second,  and  therefore  the  intensity  of  the  correspond- 
ing absorption  line,  is  hence  vanishingly  small.  By  the  dis- 
appearance of  the  first  line  to  the  right  of  the  middle  position 
v0,  the  structure  of  the  lines  as  observed  by  Imes  is  actually 
reproduced,  as  one  may  easily  recognise  ;  in  the  formula,  the 
"  middle"  of  the  line  structure  is  displaced  from  the  point  v0 

to  the  right  by  the  amount  oZFr  The  absorption  lines  group 
themselves  equidistantly  and  symmetrically  on  both  sides  of 
the  missing  "  middle,"  v  =  v0  +  Q-^J-  This  state  of  affairs 

O7T  J 

may  be  expressed  by  writing,  in  formal  agreement  with  (83), 

"  =  "''±4^7    (*-:i,8,8...)| 
where  .     (119) 


From  the  constant  interval  between  neighbouring  lines,  namely 

.     (120) 


the  moment  of  inertia  of  the  rotating  molecule  can  be  cal- 
culated with  great  accuracy.328 


CHAPTEE  IX 

The  Future 

IN  the  preceding  pages  the  author  has  attempted  to  give 
in  broad  outline  the  most  important  features  of  the 
doctrine  of  quanta,  its  origin,  its  development,  and  its 
ramifications.  If  we  now  survey  the  whole  structure,  as 
it  stands  before  us,  from  its  foundations  to  the  highest  story, 
we  cannot  avoid  a  feeling  of  admiration ;  admiration  for  the 
few  who  clear-sightedly  recognised  the  necessity  for  the  pew 
doctrine  and  fought  against  tradition,  thus  laying  the  founda- 
tions for  the  astonishing  successes  which  have  sprung  from 
the  quantum  theory  in  so  short  a  time. 

None  the  less,  no  one  who  studies  the  quantum  theory 
will  be  spared  bitter  disappointment.  For  we  must  admit 
that,  in  spite  of  a  comprehensive  formulation  of  quantum 
rules,  we  have  not  come  one  step  nearer  to  understanding 
the  heart  of  the  matter.  That  there  are  discrete  mechanical 
and  electrical  systems,  characterised  by  quantum  conditions 
and  marked  out  from  the  infinite  continuity  of  "  classically  " 
possible  states,  appears  certain.  But  where  does  the  deeper 
cause  lie,  which  brings  about  this  discontinuity  in  nature? 
Will  a  knowledge  of  the  nature  of  electricity  and  of  the  con- 
stitution of  the  electromagnetic  field  serve  to  read  the  riddle  ? 
And  even  if  we  do  not  set  ourselves  so  distant  a  goal,  there 
remains  an  abundance  of  unanswered  questions.  The 
decision  has  not  yet  been  made,  as  to  whether,  as  Planck's 
first  theory  requires,  only  quantum-allowed  states  exist  (or 
are  stable),  or  whether,  according  to  Planck's  second  formula- 
tion, the  intermediate  states  are  also  possible.  We  are  still 
completely  in  the  dark  about  the  details  of  the  absorption 
and  emission  process,  and  do  not  in  the  least  understand 
125 


126  THE  QUANTUM  THEORY 

why  the  energy  quanta  ejected  explosively  as  radiation 
should  form  themselves  into  the  trains  of  waves  which  we 
observe  far  away  from  the  atom.  Is  radiation  really  pro- 
pagated in  the  manner  claimed  by  the  classical  theory,  or 
has  it  also  a  quantum  character  ? 

Over  all  these  problems  there  hovers  at  the  present  time 
a  mysterious  obscurity.  In  spite  of  the  enormous  empirical 
and  theoretical  material  which  lies  before  us,  the  flame  of 
thought  which  shall  illumine  the  obscurity  is  still  wanting. 
Let  us  hope  that  the  day  is  not  far  distant  when  the  mighty 
labours  of  our  generation  will  be  brought  to  a  successful 
conclusion. 


Mathematical  Notes  and  References 

1 0.  Lummer  and  E.  Pringsheim,  Wiedem.  Ann.  63,  395  (1897) ; 
Verhandl.  d.  deutsch.  physikal.  Ges.  1899,  pp.  23,  215;  ibid.,  1900,  p.  163. 
Of.  also  O.  Lummer  and  E.  JahnJee,  Drudes  Ann.  3,  283  (1900),  and  O. 
Lummer,  E.  Jahnke  and  E.  Pringsheim,  Drudes  Ann.  4,  225  (1901). 

2  Of.  M.  Planck,  Vorlesungen  iiber  die  Theorie  der  Warmestrahlung 
(Leipzig  1906),  §  10. 

3  Frequency  („)  =  velocity  oflight  in  vacuo  (c)  ^ 

wave-length  in  vacuo  (A.) 

iCf.,  for  example,  M.  Planck,  Vorlesungen  iiber  Warmestrahlung 
(1906),  §  17. 

3  O.  Kirchhoff,  Gesammelte  Abhandlungen  (J.  A.  Earth,  Leipzig  1882), 
pp.  573  et  seq. ;  Berliner  Akademieberichte,  1859,  p.  216 ;  Poggend.  Ann. 
109,  275  (1860). 

6O.  Lummer  and  W.  Wien,  Wiedem.  Ann.  56,  451  (1895).  Cf.  also 
O.  Lummer  and  F.  Kurlbaum,  Verhandl.  d.  deutsch.  physikal.  Ges.  17, 
106  (1898). 

7  Of.  Note  5. 

8  L.  Boltzmann,  Wiedem.  Ann.  22,  291  (1884). 
9J.  Stefan,  Wiener  Ber.  79,  391  (1879). 

10  The  Stefan-Boltemann  Law  is  deduced  as  follows :  Let  the  energy 
of  black-body  radiation  at  the  temperature  T,  which  is  enclosed  in  a 
space  of  volume  V  having  a  movable  piston,  be  U  =  Vu,  where  u  is 
the  "  spatial "  density  of  the  radiant  energy.  The  pressure,  equal  in 
all  directions,  which  the  radiation  exerts  upon  the  piston  and  walls  is, 
according  to  electrodynamics,  p  =  $u.  If  we  supply  to  this  system  at 
the  temperature  T  (that  is,  isothermally)  an  amount  of  heat  d'Q,  then 
its  energy  increases  by  dU,  and  the  radiation  does  work  pdV  in  push- 
ing back  the  piston.  Therefore,  according  to  the  first  law  of  thermo- 
dynamics, and  owing  to  the  two  relations  above  : 

d'Q  =  dU  +  pdV  =  udV  +  Vdu  +  ™dV  =  |  udV  +  Vdu. 
According  to  the  second  law  of  thermodynamics,  -Q  must  be  a  com- 
plete differential.     Hence  the  following  relation  holds  : 
«\_    3  m  4  d     u\dT  _1 


»,'ldt*       u)dT_l 


4  fl  dw  _  u] 
3\TdT      Z^J 

127 


128  THE  QUANTUM  THEORY 

i.e.  -^  =  4^,  which,  integrated,  gives  u  =  a!" 

where  a  is  a  constant.     Now,  as  we  can  easily  see,  the  total  radiation 

00 

K  —  2  /  THvdv  is  distinguished  from  the  density  of  radiation  u  only  by 

0 

a  constant  factor  (see  M.  Planck,  Lectures  in  Radiation  (1906),  §22), 
hence  the  total  radiation  is  proportional  to  the  fourth  power  of  the 
absolute  temperature  and  this  is  the  Stefan-Boltzmann  Law. 

11  W.  Wien,  Sitzungsber.  d.  Akad.  d.  Wissensch.  Berlin,  9  Feb.  1893, 
p.  55 ;  Wiedem.  Ann.  52,  132  (1894).  Of.  also  Max  Abraham,  Theorie 
der  Elektrizitat  II,  §43  (1914);  M.  Planck,  Vorlesungen  iiber  die 
Theorie  der  Warmestrahlung  (Leipzig  1906),  pp.  68  etseq. ;  W.  Westphal, 
Verhandl.  d.  deutsch.  physikal.  Ges.  1914,  p.  93;  H.  A.  Lorentz,  Akad. 
d.  Wissensch.  Amsterdam,  18  May  1901,  p.  607. 

12 Formula  (4)  of  the  text  (Wieris  Law  of  Displacement)  may  be 
obtained  by  means  of  a  simple  dimensional  calculation,  as  L.  Hopf 
recently  showed  in  the  "  Naturwissenschaften "  (8,  109,  110  (1920)). 
We  assume  that  Kx  depends  only  on  v,  T,  and  the  velocity  of  light  c. 
The  dimensions  of  Kx  are  obtained  from  the  fact  that,  according  to  (1), 
energy 


surface  x  time 
From  this  it  follows  that 

[K,]  -  [»»«-']. 
If  we  set 

Kx  =  const.  •  vx  •  Ty  •  cz 

then,  remembering  that  T  has  the  dimensions  of  energy,  we  get 
[m£-2]  =  const.  [«-*••  my- V*  •  r *"  •  f  •  t~ Z1 

=  const.  \my  •  ?V+*  •  r*~  •»-*] 
Hence  x  =  2;        #  =  1 ;        2  =  -2 

which  gives  us,  Kx  =  const. .  ?1  .  T. 

This  relation  is  not,  however,  as  we  shall  see,  generally  valid.     In  fact, 

oo 

it  would  give  no  finite  value  for  K  =  2  \"K.vdv.     But,  according  to  the 

6 

Stefan-Boltzmann  Law  (3),  K  =  y  •  T4.  Hence  the  constant  of  Kx  may 
still  depend  on  a  dimensionless  combination  of  the  four  variables 
•y,  v,  T,  c.  If,  therefore,  we  set  const.  =f(y^Tnc^y'a)  then  the  argument 
of  the  function  /  must  have  the  dimension  0.  If,  further,  we  remember 
that 

[  BfKf*  surface  x  time 

H  =      . 


NOTES  AND  REFERENCES  129 

it  then  follows  that 

=  [t-t  .mri.fr,  t-*i  .K.t  -<T.  m-3«  .  l-**>  .  t  «•] 
=  [<  -*-«»-?+»». 


Hence  const.  =  /[(?)«".  c-  -7,]  =  < 

Therefore  K,  =  Jr.*  (j)  =  £  *.  J. 

or,  finally,  Kl>  =  3F(j} 

13  If  we  plot  K,  -  £*(£}  •••  function  of  v,  keeping  T  constant,  the 

maximum  of  this  curve  —  if  one  is  present—  lies  at  that  point  at  which 
§       =  0.    This  gives 


where  F1  is  the  differential  coefficient  of  F  with  respect  to  the  argument. 
This  equation,  in  which  only  —  occurs  as  unknown,  gives  a  definite  value 

for  ^.     In  other  words,  for  v  =  max,  it  follows  that  ^^  =  const. 

II  W.  Wien,  Wied.  Ann  58,  662  (1896). 

13  0.  Lummer  and  E.  Pringslieim,  Wied.  Ann.  63,  395  (1«97)  ;  Drude's 
Ann.  3,  159  (1900)  :  Veih.  d.  deutsch.  phys.  Ges.  1,  23  and  *15  (1899). 

The  total  radiation  emitted  per  second  from  1  cm.2  in  one  direction  is, 
by  formula  (1)  ( 

s  = 


According  to  the  Stefan-Boltzmann  Law,  S  is  nroportional  to  T4,  there- 
fore S  =  ffT*.  (The  constant  of  proportionality  a  is  related  to  the 
constant  y  occurring  in  (3)  by  the  equation  v  =  iry.)  The  absolute 
measurement  of  S  gave  the  following  values  for  a,  in  chronological  order : 

<r  -  5-45   10'12r — V!a!'t    ,"|  according  to  F.  Eurlbaum  [Wiedem.  Ann. 
Lcm/  deg.*J 

65,  746  (1898);    Verhandl.  d.  deutsch. 

physikal.  Ges.  14,  576,  792  (1912)]. 
=  5-58. 10 ~12  ti  according  to  S.  Valentiner  [Ann.  d.  Phys. 

31,  255  (1910) ;  39,  489  (1912)]. 
=  5-90 . 10 " 12  .1  according  to   W.  Gerlach  [Ann.  d.  Phys. 

38,  1  (1912)]. 


180  THE  QUANTUM  THEORY 

a-  -  5-30 . 10 ~  12TcmTde    «1  according  to  E.   Bauer  and   M.    Moulin 

[Soc.    Franc,    de    Phys.   Nr.    301,   2-3 

(1909)]. 
=  6*30   10~12  »>  according  to  Ch.  Ffry  [Bull.  Soc.  Franc. 

Phys.  4  (1909)]. 
=  6-51. 10  ~12  »>  according   to    Ch.    F&ry   and    M.   Drecq 

[Journ.  de  Phys.  (5)  1,  551  (1911)]. 
=  5*67  .  10  ~ 12  i>  according  to  G.  A.  Shakespear  [Proc.  Roy. 

Soc.  (A)  86,  180  (1911)]. 
==5-54. 10 ~12  M  according  to  W.  H.  Westphal  [Verhandl. 

d.  deutech.  physikal.  Ges.  14,  987  (1912)]. 
-  6-05. 10  "12  >,  according  to  L.  Puccianti  [Cim.  (6)  4,  31 

(1912)]. 
=  5-89. 10  ~12  ,,  according  to  Keene  [Proc.  Roy.  Soc.  (A) 

88,  49  (1913)]. 
=  5-57.10-12  „  according  to  W.  H.  Westphal  [Verhandl, 

d.  deutsch.  physikal.  Ges.  15, 897  (1913)]. 
-=  5-85 . 10  ~ 12  »,  according  to  W.  Gerlach  [Phys.  Zeitschr. 

17,  150  (1916)]. 

As  regards  Wieris  Law  of  Displacement,  the  relation  (5a)  was  tested 
and  found  to  be  confirmed.  From  Fig.  1,  in  which  E*.  is  plotted  as  a 
function  of  A.  for  different  values  of  A.,  we  see  clearly  how  the  maximum 
of  the  curve  becomes  displaced  towards  shorter  wave-lengths  as  the 
temperature  rises. 

For  the  constant  on  the  right-hand  side  of  relation  (5o)  the  measure- 
ments gave  the  following  values : 
const.  =  0-294  [cm.  deg.]  according  to  O,  Lummer  and  E.  Pringsfoim 

[Verhandl.  d.  deutsch.  physikal.  Ges.  1,  23 

and  215  (1899)]. 
=  0-292         „          according  to  F.  Paschen  [Drude's  Ann.  6,  G57 

(1901)]. 
=  0-2911       „          according  to  Coblentz  [Bull.  Bur.  of  Stand.  10, 

1  (1914)]. 

16  O.  Lummer  and  E.  Pringsheim,  Verhandl.  d.  deutsch.  physikal. 
Ges.  1,  215  (1899). 

17 F.  Paschen,  Berliner  Ber.  1899,  pp.  405,  959. 

18  M.  Planck,  Absorption  und  Emission  elektr.  Wellen  durch  Resonant. 
Sitzungsber.  d.  Berl.  Akad.  d.  Wiss.  21  March  1895,  pp.  289-301 ;  Wiedem. 
Ann.  57,  1-14  (1896).—  tJber  elektr.  Schwingungen,  welche  durch  Re- 
sonanz  erregt  und  durch  Strahlung  gedampft  werden.  Sitzungsber.  d. 
Berl.  Akad.  d.  Wiss.  20  Febr.  1896,  pp.  151-170 ;  Wiedem.  Ann.  60,  577-599 
(1897).— tTber  irreversible  Strahlungsvorgange.  (1.  Mitteilung.)  Sitzungs- 
ber. d.  Berl.  Akad.  d.  Wiss.,  4  Febr.  1897,  pp.  57-68.  (2.  Mitteilung)  ibid., 
8  July  1897,  pp.  715-717.  (3.  Mitteilung)  ibid.,  16  Dec.  1897,  pp.  1122- 
1145.  (4  Mitteilung)  ibid.,  1  July  1898,  pp.  449-476.  (5.  Mitteilung)  ibid., 
18  May  1899,  pp.  440-480.  (Supplement.)  ibid.,  9  May  1901,  pp.  544-555 ; 


NOTES  AND  REFERENCES  181 

Drudes  Ann.  1,  69-122  (1900).  (Supplement.)  Drudes  Ann.  6,  818-831 
(1901).— Entropie  und  Temperatur  strahlender  Warme.  Drudes  Ann.  1, 
719-737  (1900). 

19  In  place  of  the  mean  value,  with  respect  to  time,  of  the  energy  of  a 
single  oscillator,  we  may  use  the  spatial  mean  value  of  the  momentary 
energy  of  a  whole  system  consisting  of  very  many  oscillators. 

20  In  this  second,  more  difficult  part  of  the  calculation,  Planck  takes  his 
stand  upon  the  second  law  of  thermodynamics,  and  seeks,  from  this  view, 
to  determine  a  phase-quantity  S  of  the  oscillator,  which  possesses  the 
well-known  property  of  the  entropy,  that  it  increases  in  all  irreversible 
processes.     He  arrived  at  the  solution : 


This  function  possessed,  as  Planck  showed,  the  required  property  of  en- 
tropy, but  it  was  not  the  only  function  with  this  property.  And  in  fact 
it  appeared  later,  that  in  the  deduction  of  the  above  expression,  a  readily 
suggested  but  unjustified  supposition  had  been  made.  The  expression 
given  in  the  text,  formula  (8),  for  the  mean  energy  U  follows  from  S  by 
applying  the  second  law  in  the  form  : 


MO.  Lummer  and  E.  Pringsheim,  Verhandl.  d.  deutach.  physikal. 
Ges.  1900,  p.  163. 

22  M  .  Planck,  Verb.  d.  deutsch.  phys.  Ges.  1900,  p.  237.    It  is  of 
historic  interest  to  note  that  Planck  had  already,  in  a  somewhat  earb'er 
paper  (Verh.  d.  deutsch.  phys.  Ges.  1900,  p.  202),  arrived  at  the  true 
law  of  radiation  by  a  purely  formal  alteration  of  Wien's  formula,  which 
was  not  further  explained.     Cf.  also  Ann.  d.  Phys.  4,  553  (1901)  ;  4,  561 
(1901)  ;  6,  818  (1901)  ;  9,  629  (1902). 

23  Let  N  oscillators  be  present.     Let  the  total  energy  to  be  divided 
among  them  be  UN  =  NU.     The  "  state  "  or  phase  of  the  oscillator- 
system,  the  probability  of  which  is  to  be  calculated,  is  then  defined  by  the 
fact  that  N  oscillators  possess  the  energy  Ujt.     We  divide  UN  into  P 
energy  elements  «,  so  that 

UN  =  N  .  U  =  P*. 

The  number  of  possible  ways  of  distributing  P  balls  among  N  boxes  is, 
however, 

(N+P-  1)1 
(N  -  1)  1  P  !  ' 

This  is  therefore  the  probability  of  the  state,  which  corresponds  to  the 
distribution  of  P  energy  elements  among  N  oscillators.    P.  Ehrenfest  aud 
H.  Kamerlingh-Onnes  give  a  very  simple  deduction  of  this  formula  in 
Ann.  d.  Phys.  46,  1021  (1915). 
The  rule  mentioned  in  the  text,  which  is  due  to  Boltzmann,  states 


182  THE  QUANTUM  THEORY 

that  the  entropy  Sjt  of  the  oscillator  system  is  connected  with  the  prob- 
ability TFby  the  fundamental  relation 

Sir  =  k  log  W 
where  k  is  a  constant. 

In  this  theorem  of  Boltzmann  the  following  law  of  the  growth  of  en- 
tropy (second  law  of  thermodynamics)  is  contained  :  if  a  system  passes 
from  an  improbable  condition  into  a  more  probable  one,  then  by  this 
transition  W,  and  therefore  the  entropy  S,  increases.  If  we  here  insert  the 
value  of  W,  and,  since  N  and  P  are  very  large  numbers,  use  Stirling's 
approximation  formula 

\oge(Nl)=N(logeN-  1) 

then,  if  we  set  for  P,  N  —  ,  we  get  by  an  easy  calculation 


and  hence  the  entropy  S  of  one  oscillator  becomes  : 


But  according  to  the  Second  Law  (see  note  20) 
dS       1 


If  we  carry  out  the  differentiation  on  the  left-hand  side,  and  solve  the  re- 
sulting relation  between  U,  T,  and  e,  with  respect  to  U,  we  get  the  ex- 
pression (9)  of  the  text. 

2<  Of.  the  paper  by  Ehrenfest  and  Katnerlingh-Onnes  cited  in  the 
previous  note. 

2S  This  law  is  essentially  identical  with  Boltzmann'  's  H-  Theorem.  Of. 
L.  Boltztnann,  Vorlesungen  iiber  Gastheorie  Bd.  I,  p.  38  (1896)  ;  Sit- 
zungsber.  d.  Wiener  Akad.  d.  Wiss.  (II)  76,  373  (1877).  Of.  also  P. 
Ehrenfest,  Phys.  Zeitschr.  15,  657  (1914). 

36  H.  Rubens  and  F.  Kurlhaum,  Sitzungsber.  d.  Berl.  Akad.  d.  Wiss. 
1900,  p.  929  ;  Ann.  d.  Phys.  4,  649  (1901). 

27  F.  Paschen,  Ann.  d.  Phys.  4,  277  (1901). 

28  L.  Holborn  and  8.  Valentiner,  Ann.  d.  Phys.  22,  1  (1907)  ;  Coblentz, 
Physical  Review,  31,  317  (1910)  ;  E.  Baisch,  Ann.  d.  Phys.  35,  543  (1911)  ; 
E.  Warburg,  O.  Leithttuser,  E.  Hupka  and  C.  Milller,  Ann.  d.  Phys.  40, 
609  (1913)  ;  E.  Warburg  and  C.  Milller,  Ann.  d.  Phys.  48,  410  (1915). 

»  W.  Nernst  and  Th.  Wulf,  Ber.  d.  deutsch.  phys.  Ges.  21,  294  (1919). 
80  Lord  RayUigh,  Phil.  Mag.  49,  539  (1900). 

31  The  "  Stefan-Boltzmann  constant  of  total  radiation"  <r,  introduced 
in  note  15,  has  therefore  the  value 


NOTES  AND  REFERENCES  133 

32  In  order  to  determine  the  constants  h  and  k  which  occur  in  the 
radiation  formula,  we  can,  instead  of  using  the  equation :  \max .  T  = 
const.,  compare  other  relations  with  the  measurement  of  the  total 
radiation.  For  example,  we  can  proceed  as  follows :  At  a  constant 
temperature  T  we  measure  the  ratio  of  the  intensity  of  radiation  for  two 
different  wave-lengths  \j  and  \^  (isothermal  method).  Now  this  ratio 
is,  according  to  (15) 


•&*»      \A1/          _C_  K 

e*lT  -  1 

From  this  relation,  since  everything  excepting  C  is  known,  C,  that  is, 
-  may  be  calculated.  Another  method  is  the  following  :  we  measure  for 

a  fixed  wave-length  x  the  ratio  of  the  intensity  of  radiation  at  two 
different  temperatures  Tl  and  T2  (isochromatic  method).  Then  it  follows 
that 


This  is  a  relation  from  which  C,  that  is,—-  can  again  be  calculated. 

With  the  help  of  these  methods,  the  researches,  for  example,  of 
Warburg  and  his  co-workers  cited  in  note  28  have  yielded  values  for 

C  =  ^  which  lie  in  close  proximity  to  C  =  1-430.    This  value  was  taken 

by  Nernst  and  Wulf  (see  note  29)  for  their  critical  investigation. 

For  the  constant  of  Wien's  Law  of  Displacement  in  the  form  \inax  .  T 
=  b  we  would  accordingly  get  from  (16) : 


4-9651 

a  value  smaller,  therefore,  than  that  given  by  direct  measurement  (see 
no'e  15).  Whether  Warburg's  value,  C  =  1-430,  or  the  measured  values 
of  &(>  0-29)  or  both,  are  seriously  affected  by  experimental  error,  or 
whether  after  all — as  Nernst  and  Wulf  maintain — Planck's  formula  is 
not  right,  must  be  left  for  the  future  to  decide. 

38  M.  Planck,  Ann.  d.  Phys.  4,  553  (1901). 

84 If  we  apply  Boltzmann's  relation  S  =  k  log  W  (quoted  in  note  15), 
which  connects  the  entropy  S  with  the  probability  of  state  W,  to  one 
gramme-molecule  of  an  ideal  gas,  then  by  calculating  the  probability  of 
a  certain  state,  i.e.  a  certain  distribution  of  velocities  among  the 
molecules,  we  arrive  at  the  following  value  for  the  entropy  of  the  gas 

S  «  kN(l  log*  U  +  log  V)  +  const. 


184  THE  QUANTUM  THEORY 

(Of.,  for  example,  M.  Planck,  Lectures  on  the  Theory  of  Radiation 
(1906),  §  143.)  Here  N  is  the  number  of  molecules  in  a  gramme- 
molecule  (Avogadro's  number),  U  the  energy,  V  the  volume  of  the  gas. 
Now,  according  to  the  Second  Law  of  Thermodynamics, 

jo      dU  + 


must  be  a  complete  differential,  where  p  and  T  denote  pressure  and 
temperature  of  the  gas.     Hence  the  relation 


\dVju~  T 

must  hold.    This  gives 

kV=  P    ie  p_kNT 

If  we  compare  this  with  the  equation  of  state  of  an  ideal  gas  in  thermo- 
dynamics, p  =  ~£,  we  get  for  the  absolute  gas  constant  R  the  value 
B  =  kN 

from  which  formula  (19)  of  the  text  follows. 

88  M.  Planck,  Ann.  d.  Phys.  4,  564-566  (1901). 

38  Compare,  for  example,  the  table  of  the  values  of  Avogadro's  number 
given  in  the  report  of  J.  Perrin  at  the  Solvay  Congress  in  Brussels 
(1911).  [A.  Eucken,  Die  Theorie  der  Strahlung  und  der  Quanten. 
Abhandlungen  der  Bunsen-Gesellschaft  Nr.  7,  Wilh.  Knapp,  Halle  1914.] 

37 R.  A.  Millikan,  Phil.  Mag.  (6)  34,  13  (1917). 

Mlbid.,  from  the  values  given  by  Millikan  for  the  electronic  charge 
e  =  4-774  x  10-10  (electrostatic  units)  and  from  the  electrochemical 
constant  F  =  969-4  .  2'999  . 1010  electrostatic  units,  there  follows  for 
Avogadro's  number  the  value  N  =  6-0617 . 1023. 

39 Of.,  for  example,  W.  Gibbs'  Elements  of  Statistical  Mechanics, 
Chapter  V. 

WThe  term  "mean  value"  may  be  taken  as  referring  to  time  or  to 
space.  If  we  select  a  definite  atom,  and  follow  it  a  long  time  upon  its 
zig-zag  path,  and  from  the  mean  of  the  values  which  its  kinetic  energy 
assumes  in  the  course  of  time,  we  get  the  "  time-mean."  If,  on  the 
other  hand,  we  select  a  large  number  of  identical  atoms  of  the  gas  at  a 
particular  instant  and  again  form  the  mean  of  the  values  of  the  kinetic 
energies  which  these  atoms  possess  at  the  instant  in  question,  we  get 
the  "  space-mean." 

*1  If  x  is  the  elongation  of  the  oscillator  (electron)  vibrating  with  the 
natural  frequency,  then  x  =  A  sin  (2mrf),  where  A  is  the  amplitude  and 
t  the  time  ;  the  mean  kinetic  energy  becomes 


NOTES  AND  REFERENCES  135 

The  mean  potential  energy  is  : 
V 


Hence,  as  stated,  L  =  V:  i.e.  the  mean  kinetic  energy  =  the  mean 
potential  energy. 

42  /.  H.  Jeans,  Phil.  Mag.  10,  91  (1905). 

MH.  A.  Lorentz,  Proc.  Kon.  Akad.  v.  Wet.,  Amsterdam  1903,  p.  666. 
—The  theory  of  electrons  (Teubner,  Leipzig  1909),  Oh.  II. 

444.  Einstein  and  L.  Hopf,  Ann.  d.  Phys.  33,  1105  (1910). 

UA.  D.  Fokker,  Ann.  d.  Phys.  43,  810  (1914). 

46  M.  Planck,  Ber.  d.  Berl.  Akad.  d.  Wiss.,  8  July  1915,  p.  512. 

47  H.  A.  Lorentz.     Die  Theorie  d.  Strahlung  u.  d.  Quanten  ;  Abhand- 
lungen  der   Deutschen   Bunsen-Gesellschaft.      Nr.   7.   v.   A.  Eitcken. 
Halle,  W.  Knapp  1914  pp.  10  et  seq. 

48  By  a  suitable  modification  of  classical  statistics  in  the  sense  of  the 
quantum  theory,  we  can  obtain  the  expression  (9)  for  the  mean  energy 
of  an  oscillator  in  the  following  manner  which  is  worthy  of  notice. 
Let  a  number  N  of  similar  oscillators  with  the  most  varied  values  for 
the  energy  be  given.     We  require  to  find  how  great  is  the  probability 
w,  that  an  oscillator  possess  a  certain  energy  value  U;  or,  otherwise 
expressed,  how  many  of  the  N  oscillators  possess  the  energy  U.    In 
order  to  answer  this  question,  we  find  it  best  to  take  first  of  all  the 
standpoint  of  Oibbs'    statistical    mechanics,  that  is,   of  "  classical  " 
statistics.     In  place  of  the  special  case  in  question,  namely,  that  of  the 
linear  oscillator,  let  us  consider  at  once  quite  generally  a  system  of  / 
degrees  of  freedom,  and  characterise  it  by  /  generalised  co-ordinates 
2i2z  •  •  •  ^  and  by  the  corresponding  impulses  or  momenta  2*1  .Pa  •  .  .  Pf. 
(Here,   the  impulse  pi  is  thus  defined  :  form  the  kinetic   energy  of 

the  system  as  a  function  of  the  generalised  velocities  qi  —   J£-,  then 

OT-     .  <** 

pi  =  -~r.  \     In  particular,  the  linear  oscillator  (vibrating  electron)  will 

°qi' 

be  described  by  a  co-ordinate  q,  namely,  the  elongation  of  the  electron, 
and  the  impulse  p  =  m  -%.  In  general,  therefore,  2/  quantities  are 

necessary  in  order  to  define  completely  the  momentary  state  of  a 
system.  Hence  we  can  represent  this  momentary  state  by  a  point 
("  phase-point  ")  in  the  2/-  dimensional  space  in  which  9i  .  .  .  Pf  (of  the 
"  phase-space  ")  are  co-ordinates. 

We  now  consider  a  number  N  of  similar  systems  of  this  kind, 
which  are  in  thermodynamic  equilibrium  with  a  very  large  reservoir 
at  the  temperature  T.  Then  the  probability  that  the  co-ordinatea 
and  impulses  lie  in  the  small  intervals  q1  .  .  .  gt  +  dqv  etc.,  and 
Pi  .  .  .  Pi  +  rfpj,  etc.,  that  is,  that  the  "phase-point"  of  the  system  lie 
in  the  element  dn  =  dq^dq^  .  .  .  dqj,  dp^pt  .  .  .  dpf  of  the  phase- 
space  is,  according  to  Oibbs, 


136  THE  QUANTUM  THEORY 

Here  E  is  the  energy  of  the  system,  and  k  is  the  constant  defined  in 
(19).  The  integration  in  the  denominator  is  to  be  taken  over  all 
possible  values  of  the  2/  quantities  q1  .  .  .  pf,  or,  as  we  may  say,  over 
all  possible  "  phases,"  or  over  the  whole  region  of  the  phase-space 
concerned. 

Among  the  N  systems  there  are  then  Nw,  whose  phase-points  lie  in 
the  element  dfi  of  the  phase-space.  This  is  therefore  a  "  distribution  " 
of  the  N  systems  over  the  phase-space.  This  distribution  is  called 
Canonical ;  it  represents  a  generalisation  of  Maxwell's  familiar  law  of 
distribution  of  velocities  which  may  be  deduced  from  it  by  special- 
ising it  for  the  case  of  the  gas  atom,  that  is, by  setting/  =3. 

The  sum  of  all  probabilities  is  naturally  1.  Indeed,  it  is  at  once 
clear  that 


L-> 


For  the  mean  value  of  the  energy  E  we  get 

J-M.  -/*'**». 
/."»*> 

If  we  apply  this  equation  to  the  linear  oscillator  we  get 

/  f     -— 
TTJjUe   Wgqdp 

ffe-^dp 
N°W'  <7  -!'  +      -0MV 


If  we  introduce  the  auxiliary  variables  {  and  n,  defined  by 
j£  =  T«W2m 

j  n  =     ,JL,  and  hence  dqdp  =  —d£dn 
*•         fj'2m  TV 

we  get 

U  =  f2  +  T? 
and,  therefore,  it  suggests  itself  to  us  to  write 


f|= 

\rj= 


where  <f>  is  a  parametric  angle.    If  we  interpret  {  and  rj  as  Cartesian  co- 
ordinates of  a  point  in  the  plane,  then  JU  and  <p  are  the  polar  co-ordinates 


NOTES  AND  REFERENCES  137 

of  this  point.     The  element  of  surface  d£drj  is  written  in  polar  co-ordinates, 
as  we  know,  thus 


hence 


Hence 


dqdp  =  — 


r       r     -£ 
J        Jta    M 

tr=0«*>  =  0 


in  agreement  with  (24).     This  is  the  standpoint  of  classical  statistics. 

The  quantum  statistics  of  the  oscillator  may  be  immediately  deduced 
from  this,  if  we  elaborate  the  canonical  law  of  distribution 


i  - 

Je 


— 

kTdqdp 


in  a  suitable  manner 

If  we  here  again  introduce  dqdp  =  —-dUdf,  and  integrate  with  respect 
to  $,  we  get 

_£ 

wu=  e   *rdU 

Je'^dU 

as  the  probability  that  the  energy  of  the  oscillators  lies  between  U  and 
U  H-  dU. 

Now  the  quantum  theory  demands  that  the  energy  U  shall  assume  only 
the  discrete  values  U0,  Ult  U0,  .  .  .  Un-  The  transition  may  best  be 
effected  by  laying  down  the  condition :  E  shall  only  be  able  to  assume  the 
values  contained  in  the  narrow  intervals  between  U0  and  U0  +  a,  U^  and 
I7j  +  o,  and  generally  Un  and  Un  +  o.  Then  dU  =  a,  and  the  integral  in 
the  denominator  changes  into  a  sum.  Thus  it  follows  that 


n  n 

thus  a  is  eliminated ;  if  we  now  proceed  to  the  limit  a  =  0,  w  remains 


138  THE  QUANTUM  THEORY 

unaltered.  Hence  wn  is  the  canonical  distribution  function  generalised 
for  quantum  conditions,  and  hence,  among  N  oscillators,  Nwn  have  an 
energy  of  the  value  Un. 

We  now  get  for  the  mean  energy 

V          *« 

2ft*  v 


Now,  according  to  the  first  form  of  the  quantum  theory, 

Un  =  tie  =  nhv         (n  =  0,  1,  2,  8  .  .  .  oo  )". 
Therefore 


u  = 

2 


V 

o  S, 


o 
If  we  set  fc5>,  for  convenience,  =  x,  then 


Further, 


from  which  we  get 


in  agreement  with  (9) 

The  canonical  distribution  may  be  still  further  generalised  by  the  intro- 
duction of  certain  "weight  factors,"  which  are  intended  to  express  the 
fact  that  the  individual  quantum  states  of  the  system  considered  have, 
a  priori,  different  probabilities.  This  happens,  for  example,  if  each  quantum 
state  may  be  realised  in  different  ways,  and  if  the  number  of  these  possi- 
bilities of  realisation  is  different  for  the  different  quantum  states.  Then, 

the  different  states  will  have  different  "weights,"  and  a  "weight  factor" 

vn 

pn  has  to  be  included  in  the  exper  mental  function  e  ~*r  so  that  the  can- 
onical distribution  function  assumes  the  form 


-  =  C  .  f^ff' 
Here  C  depends  on  the  temperature  ;  pn,  on  the  other  hand,  doe«  not. 


NOTES  AND  REFERENCES  189 

194.  Einstein,  Ann.  d.  Phys.  17, 132  (1905) ;  20, 199  (1906) ;  Verhandl. 
d.  deutsch.  physikal.  Ges.  11,  482  (1909) ;  Bericht  Einstein  auf  dem 
Solvay-Kongress  in  Brussels  1911 ;  cf.  A.  Eucken,  Die  Theorie  der 
Strahlung  und  der  Quanten ;  Abhandl.  d.  deutsch.  Bunsen-Gesellschaft, 
Nr.  7  (Halle,  W.  Knapp  1914),  pp.  330  et  seq.  Cf.  also  W.  Wien, 
Vorlesungen  iiber  neuere  Probleme  der  theoretischen  Physik  (Teubner, 
Leipzig  and  Berlin  1913),  4.  Vorlesung.  H.  A.  Lorentz,  Les  theories 
statistiques  en  thermodynamique  (Teubner,  Leipzig  and  Berlin  1916), 
§§  42  et  seq. 

SO  A.  Einstein,  Ann.  d.  Phys.  17,  132  (1905). 

31  A.  Einstein,  Phys.  Zeitschr.  10,  185  (1909). 

52  This  formula  may  be  deduced  as  follows  :  Firstly,  from  e  =  E  -  E 
the  frequently  used  relation 

?  =  W  -  2E  .  IS  +  (E)*  =  W  -  (IT)2 

follows.  In  order  now  to  calculate  the  two  quantities  E*  (mean  of  the 
squares  of  the  energy)  and  (E)*  (square  of  the  mean  energy),  which 
are  known  to  differ  from  each  other  in  general,  we  do  best  to  take  the 
standpoint  of  Gibbs'  statistical  mechanics  (see  note  48).  According  to 
this,  the  probability  that  the  co-ordinates  and  impulses  lie  in  the  small 
intervals  ql  .  .  .  ql  +  dqlt  etc.,  p^  .  .  .  p1  +  dplt  etc,  that  is,  that  the 
"phase-point"  lies  in  the  element  dqtdq2  .  .  .  dqfdp^dp^  .  .  .  dp,  =  dn 
of  the  "  phase-space  "  : 


Then  the  mean  of  the  energy  follows  in  the  usual  way  : 


Likewise, 
We  then  form 

dE 


140  THE  QUANTUM  THEORY 

Therefore, 

dE 
F-Wjy 

We  also  arrive  at  the  same  formula,  if  instead  of  the  classical 
canonical  distribution  function,  we  start  from  the  quantum  distribution 
function 


68  The  mean  energy  of  radiation  of  frequency  v  in  the  volume  v  is 
=  vuvdv,  where  the  monochromatic  density  of  radiation  is 


if  Planck's  Law  is  taken  as  the  basis.     (Of.,  for  example,  M.  Planck, 
Lectures  on  the  Theory  of  Radiation,  Engl.  Transl.) 

According  to  formula  (28)  deduced  in  the  previous  note,  it  therefore 
follows  that 


If  we  eliminate  T  on  the  right-hand  side  by  substituting  for  e**  its 
value  1  +  8vhvS,  it  follows  that 

C'Uy 


uvvdv  , 


The  second  term  on  the  right  is  required  by  the  Undulatory  Theory 
for  at  each  point  of  the  volume  v  the  most  varied  trains  of  waves  of 
radiation  cross  one  another's  paths  with  every  possible  amplitude  and 
phase.  The  interference  of  all  these  waves  thus  generates  at  the  point 
considered  an  intensity,  which  varies  continually,  and  hence  the  energy 
of  the  volume  v  also  varies.  If  we  calculate  the  mean  of  the  square 
of  the  energy,  i.e.  «a,  wo  find  precisely  the  second  term  of  the  above 
formula.  (Of.,  for  example,  H.  A.  Lorents,  Les  theories  statistiques  en 
thermodynamique  (Teubner,  Leipzig  and  Berlin),  1916,  pp.  114  et  seq,) 

The  first  term  is  not,  however,  explained  by  the  classical  undulatory 
theory.  On  the  other  hand,  it  becomes  endowed  with  meaning  if  we 
suppose  that  the  radiant  energy  consists  of  a  certain  whole  number 


NOTES  AND  REFERENCES  141 

(n)  of  finite  energy  complexes  of  the  value  hv.  For  then  E  =  n  •  hy, 
and  therefore  E  =  n  •  hv,  where  n  is  the  mean  about  which  the  number 
n  varies.  If  5  =  n  -_  n  be  the  variation  of  the  number  n,  then  it 
follows  that  e  =  E  -  E  =  Shy^  where  «2  =  S2  •  feV.  But,  according  to 
a  well-known  law  of  statistics,  S2  =  n.  (Cf.,  for  example,  H.  A,  Lorentz, 
loc.  cit.,  §§  26  and  27.)  Hence  ?  =  nfeV  =  ~E'hv.  This  is  exactly  the 
first  term  in  the  above  formula. 

MA.  Einstein,  Ann.  d.  Phys.  17,  144  (1905). 

95  J.  J.  Thomson,  Conduction  of  Electricity  through  Gases. 

96.4.  Einstein,  Ann.  d.  Phys.  17,  147  (1905). 

97  Of.  R.  Pohl  and  P.  Pringsheim,  Die  lichtelektrischen   Erschein- 
ungen.  Sammlung  Vieweg  Heft  1  (Braunschweig  1914). 

98  A.  Einstein,  Ann.  d.  Phys.  17,  145  (1905). 

99  R.  A.  Millikan,  Phys.  Zeitschr.  17,  217  (1916). 

60  According  to  Pohl  and  Pringsheim,  we  have  to  distinguish  between 
the  normal  and  the  selective  photo-effect :   in  the  case  of  the  normal 
effect  the  number  of  electrons  torn  off  (per  calorie  of  the  Jight-energy 
absorbed)  is  independent  of  the  orientation  of  the  electrical  vector  of  the 
light-wave,  and  increases,  starting  from  an  upper  limit  of  the  wave- 
length, in  general  uniformly  as  the  wave-length  decreases.     In  the  case 
of  the  selective  effect,  on  the  other  hand,  which  only  appears  when  the 
electrical  vector  of  the  light-wave  possesses  a  component  vertical  to  the 
metallic  surface,  the  number  of  electrons  torn  off  (per  calorie  of  light- 
energy  absorbed)  shows  a  decided  maximum  at  a  definite  wave-length. 

61  Oh.  Barkla,  Phil.  Mag.  7,  543,  812;  15,  218.     Jahrb.  d.  Radioak- 
tivitat  u.  Elektronik,  5,  p.  239, 1908.— Ch.  Barkla  and  Sadler,  Phil.  Mag. 
17,  739.— Ch.  Barkla,  Jahrb.  d.  Radioaktivitat  u.  Elektronik,  1910,  p. 
12.— if.  de  Broglie,  G.  R.  25  May  and  15  June  1914,  p.  1785.— Ch.  Barkla, 
Phil.  Mag.  16,   550.— E.   Wagner.  Ann.  d.  Phys.  46,  868  (1915);   Sit- 
zungsber.  d.  bayer.  Akad.  1916,  p.  39. 

62  D.  L.  Webster,  Proc.  Americ.  Acad.  2,  90(1916);  Physic.  Review,  7, 
587  (1916). 

63  E.  Wagner,  Ann.  d.  Phys.  46,  868  (1915). 

64  Of.,  for  example,  E.  Wagner,  Phys.  Zeitschr.  18,  443  (1917).     The 
value  that  Wagner  calculates  for  h  is :  h  =  6-62 .10-27. 

65  W.  Duane  and  F.  L.  Hunt,  Physic.  Review,  6,  166  (1915). 
86.4.  W.  Hull  and  If.  Rice,  Proc.  Americ.  Acad.  2,  265  (1916). 

67  E.  Wagner,  Phys.  Zeitschr.  18,  440  et  seq.  (1917) ;  Ann.  d.  Phys.  57, 
401  (1918). 

68  F.  Dessauer  and  E.  Back,  Ber.  d.  deutsch.  physikal.  Qes.  21,  168 
(1919). 

69  J.  Franck  and  G.  Hertz,  Verhandl.  d.  deutsch.  physikal.   Ges.  16, 
512  (1914). 

70  The  critical  potential  measured  by  Franck  and  Hertz  amounted  to 

V=  4-9  volts  =  —  electrostatic  units,  and  therefore  the  critical  energy 

300 
of  the  electron  is 

v  _  4-774. 10  -10.  4-9 
300 


142  THE  QUANTUM  THEORY 

The  wave-length  A  of  the  mercury  line  emitted  is 
\  =  25364  =  2-536.10-5. 
Hence  we  must  get 

17     j.e   •      i.      «V*      4-774.  10 -10.  4-9.  2-536. 10 -5 
,7=  fc-,  ,.e.  h  =  —  =   -  3.102-3^r 

=  6-59.10-27 

and  this  is  in  good  agreement  with  the  results  of  other  measurements. 

71  Cf.,  for  example,  J.  Stark,  Prinzipien  der  Atomdynamik  II.  (S. 
Hirzel,  Leipzig  1911),  Chs.  IV  and  V. 

72  J.  Stark,  Ber.  d.  deutsch.  phys.  Ges.  10,  713  (1908) ;  Phys.  Zeitschr. 
8,  913  (1907) ;  9,  767  (1908). 

Canal-rays  are  positively  charged  particles  of  matter,  which  move  in  a 
vacuum  tube  in  the  direction :  anode  to  cathode ;  the  latter  is  pierced 
with  holes  through  which  the  canal-rays  pass  into  the  space  behind  the 
cathode.  If  we  generate  such  canal-rays  in  a  vacuum  tube  rilled  with 
hydrogen,  we  find  that  the  series  lines  of  hydrogen  are  emitted.  Now, 
if  we  observe  this  emission  spectroscopically  "  from  the  front,"  that  is,  so 
that  the  canal-rays  are  moving  towards  the  observer,  we  see,  firstly,  at 
its  usual  place  in  the  spectrum,  the  sharp  series  line  (line  of  rest,  "in- 
tensity of  rest ") ;  secondly,  we  see  displaced  towards  the  violet,  a 
broadened  strip  (line  of  motion,  "intensity  of  motion"  or  "dynamic 
intensity").  These  lines  represent  the  series  line  emitted  by  the 
moving  canal-ray  particles,  which  is  displaced  towards  the  region  of 
higher  frequencies  on  account  of  the  Doppler  effect.  Since  the  canal-rays 
do  not  possess  a  single  uniform  velocity,  and  since  particles  with  all 
possible  velocities  occur,  the  displaced  strip  is  not  sharp,  but  softened 
and  broadened.  The  "  intensity  at  rest  "  is  therefore  emitted  when  the 
quickly  moving  canal  particles  strike  "  resting "  molecules,  i.e.  gas- 
molecules  which  are  moving  comparatively  slowly  and  irregularly,  and 
excite  these  to  emit  the  series  lines.  The  "  intensity  of  motion,"  on  the 
other  hand,  is  excited  by  the  unidirectionally  moving  canal  particles 
themselves,  when  they  hit  gas-molecules. 

Now,  it  is  very  remarkable  that  the  interval  between  the  intensity  of 
rest  and  that  of  motion  is  not  filled  in,  but  that  the  emission  of  the  in- 
tensity of  motion  becomes  observable  only  above  a  certain  velocity. 
Stark  interpreted  this  fact  in  terms  of  the  light-quantum  hypothesis 
thus :  If  %mvz  is  the  kinetic  energy  of  a  canal-ray  particle,  and  if  the 
fraction  a£rau2(a  >  1)  is  transformed  into  a  light-quantum  hv  upon 
collision  with  a  gas-molecule,  then  we  must  have  h»  <  -me2 ;  that  is,  the 
spectral  line  of  frequency  v  can  be  generated  only  by  canal-rays,  the 

Telocity  of  which  >A/?5 
\  can 

The  proportionality  between  the  critical  velocity  and  *Jv  has  been 
fairly  well  borne  out. 

It  should  be  remarked  here  that  J.  Stark  has  lately  abandoned  the 
theory  of  light-quanta.  (Cf.  J.  Stark,  Verh.  d.  deutsch.  physik.  Ges.  16, 
304  (1904) ;  18,  42  (1916).) 


NOTES  AND  REFERENCES  148 

73  J.  Stark,  Phys.  Zeitschr.  9,  85,  356  (1908).—  J.  Stark  and  W. 
Steubing,  Phys.  Zeitschr.  9,  481  (1908).—  J.  Stark,  Phys.  Zeitschr.  9, 
889  (1908). 

In  these  papers  J.  Stark  defends  the  view  that  the  band-spectra  are 
emitted  when  a  "valency  electron"  belonging  to  the  atom  or  molecule 
is  pushed  out  of  its  normal  position  and  then  returns  again  to  its  initial 
position,  counterbalancing  the  work  done  in  displacement.  If  the 
energy  of  deformation  (valency  energy)  E  is  changed  into  a  light- 

quantum,  then  we  must  have  hv  =  E,  i.e.  v  ^-  —      All  lines  of  the 

^  7t 

ET 

band  must  therefore  lie  below  the  edge  v  =  _  .     If  the  valency  energy 

E  is  changed  by  chemical  processes,  the  band-spectrum  must  be  dis- 
placed accordingly. 

74,1  Einstein,  Ann.  d.  Phys.  17,  148  (1905). 

75  J.  Stark,  Phys.  Zeitschr.  9,  889  (1908)  ;  Ann.  d.  Phys.  38,  467  (1912). 

The  fundamental  law  of  photochemical  decomposition  enunciated  by 
Stark  and  Einstein  states  :  If  a  molecule  dissociates  at  all  owing  to  j,he 
absorption  of  radiation  of  frequency  v,  then  it  will  absorb  an  amount  of 
energy  hv  when  it  dissociates.  This  energy,  therefore,  represents  the 
heat  of  reaction,  which  will  be  set  free  upon  recombination  of  the 
products  of  decomposition. 

This  law  was  later  deduced  by  A.  Einstein  for  the  range  of  validity  of 
Wierfs  Law  of  Radiation  without  the  assistance  of  the  light-quantum 
hypothesis,  by  purely  thermodynamical  methods.  (Of.  Ann.  d.  Phys. 
37,  832  (1912),  and  38,  881  (1912).) 

nE.  Warburg,  Ber.  d.  Berl.  Akad.  d.  Wiss.  1911,  p.  746;  1913,  p. 
644  ;  1914,  p.  872  ;  1915,  p.  230  ;  1916,  p.  314  ;  1918,  pp.  300,  1228.  Cf. 
also  "  Naturwissenschaften,"  5,  489  (1917). 

77  H.  A.  Lorentz,  Phys.  Zeitschr.  11,  1250  (1910). 

78  M.  Planck,  Ber.  d.  deutsch.  physikal.  Ges.  13,  138  (1911);  Ann.  d. 
Phys.  37,  6i2  (1912). 

79  On  account  of  the  continuous  (classical)  absorption,  all  energy  values 
of  the  oscillator  in  an  elementary  region,  say  between  n«  and  (n  +  l)e, 
are  equally  probable.     The  mean  energy  in  the  nth  elementary  region 
is,  therefore, 


Prom  the  canonical  law  of  distribution  extended  in  the  sense  of  the 
quantum  theory,  it  then  follows  that 


144  THE  QUANTUM  THEORY 


0 
(cf.  note  48).     If  we  further  set  e  =  hy  it  follows  that 


In  place  of  relation  (7)  of  the  text  we  get  here 


and  this  leads  to  Planck's  Law  of  Radiation. 

80  M.  Planck,  Sitzungsber.  d.  Kgl.  Preuss.  Akad.   d.   Wiss.  3  April, 
1913,  p.  350;  ibid.,  30  July  1914,  p.  918;  ibid.,  8  July  1915,  p.  512. 

81  A.  Einstein  and  O.  Stern,  Ann.  d.  Phys.  40,  551  (1913). 

82  W.  Nernst,  Verhandl.  d.  deutsch.  physikal.  Ges.  18,  83  (1916). 

83  F.  Richarz,  Wiedem.  Ann.  52,  410  (1894J. 

84  Report  by  P.   Langevin  at  the  Solvay  Congress  in  Brussels,  1911. 
Cf.  A.  Eucken,  Die  Theorie  der  Strahlung  und  der  Quanten.     Abhandl.  d. 
deutsch.  Bunsen-Ges.,  Nr.  7  (W.  Knapp,  Halle  1914),  pp.  318  et  seq. 

8M.  Einstein  and  W.  J.  de  Haas,  Verhandl.  d.  deutsch.  physikal.  Ges. 
17,  152,  203,  420  (1915).—  A.  Einstein  ibid.,  18,  173  (1916).—  W.  J.  de 
Haas,  ibid.,  18,  423  (1916). 

86  E.  Beck,  Ann.  d.  Phys.  60,  109  (1919). 

87  Report  by  Planck  at  the  Solvay  Congress  in  Brussels,  1911.      See 
A.  Eucken,  Die  Theorie  der  Strahlung  und  der  Quanten.    Abhandl.  d. 
deutsoh.  Bunsen-Ges.,  Nr.  7  (W.  Knapp,  Halle  1914),  p.  77. 

88  If  q  is  the  elongation  of  a  linearly  vibrating  electron  of  mass  ra  (os- 
cillator) and  v  its  period  of  oscillation,  then  the  energy  of  this  configur- 
ation is 


The  first  term  represents  the  kinetic  and  the  second  the  potential  energy. 
Now  the  impulse  (the  momentum)  is  p  =  m~Ti-    Therefore,  we  may  write 


i.e. 


NOTES  AND  REFERENCES  145 

The  curves  U  =  const.,  that  is,  those  curves  in  the  phase-plane,  which 
correspond  to  the  states  of  constant  energy  of  the  oscillator,  are  therefore 
ellipses  with  the  semi-axes 


For  a  definite  value  of  U  we  get  a  completely  definite  ellipse.  The 
"  phase-point  "  of  the  oscillators  would  continually  revolve  in  this  ellipse, 
if  the  electron,  without  emitting  or  absorbing,  were  to  execute  pure  har- 
monic oscillations  :  for  then  its  energy  would  remain  permanently  constant. 
If  we  allow  U  to  vary  continuously,  i.e.  if  we  give  it  other  and  again 
other  values  in  continuous  succession,  we  get  an  unlimited  manifold  of 
concentric  ellipses. 

The  quantum  theory,  as  formulated  in  (30)  in  the  text,  selects  from  this 
infinite  manifold  a  discrete  set  of  ellipses,  and  distinguishes  them  as  the 
"quantised"  ellipses  which  correspond  to  the  "  characteristic  states  "  of 
the  oscillator.  To  these  belong  the  "quantum  energy-  values  "  U0,  Ult 
Ua  .  .  .  Dn. 

Now  the  nth  ellipse  encloses  an  area  nh.  The  area  of  the  nth  ellipse 
is,  however, 


hence  we  must  have 


—  =  nh    i.e.     Un  = 


that  is,  in  the  nth  quantum  state  the  oscillator  possesses  an  amount  of 
energy  nt  =  nhv. 

tOA.  Sommerfeld,  Phys.  Zeitschr.  12,  1057  (1911).—  Report  by  A. 
Sommerfeld  at  the  Solvay  Congress  in  Brussels,  1911.  Cf.  A.  Eucken, 
Die  Theorie  der  Strahlung  und  der  Quanten.  Abhandl.  d.  deutsch. 
Bunsen-Ges.,  Nr.  7  (W.  Knapp,  Halle  1914),  p.  252. 

80  Report  by  Sommerfeld  at  the  Solvay  Congress,  1911. 

91  A.  Sommerfeld  and  P.  Debye,  Ann.  d.  Phys.  41,  873  (1913). 

92  Cf.,  for  example,  the  recent  summary  by  E.   Schrddinger,   Der 
Energieinhalt  der  Festkorper  im  Lichte  der  neueren  Forschung.      Phys. 
Zeitschr.  20,  420,  450,  474  (1919).    A  complete  set  of  references  accom- 
panies this  account. 

93  One  gramme-atom  of  a  substance,  the  atomic  weight  of  which  is  a, 
is  defined  as  the  quantity  a  grammes  of  the  substance.     For  example, 
one  gramme-atom  of  copper  is  equal  to  63'57  grammes  of  copper,  since 
63-57  is  the  atomic  weight  of  copper.      Exactly    analogous   is    the 
definition  of  the  gramme-  molecule  (also  called  "  mol").     One  gramme- 
molecule  of  oxygen  is  32  grammes  of  oxygen,  for  the  molecular  weight 
of  oxygen  (diatomic)  is  32. 

If  c  is  the  specific  heat  of  a  substance  of  atomic  weight  a,  it  signifies 
that  one  gramme  of  the  substance  requires  an  amount  of  heat  c  to  raise 
its  temperature  by  1°  C.     Hence  we  must  communicate  to  a  gramme-atom 
10 


146  THE  QUANTUM  THEORY 

of  the  substance,  i.e.  to  a  grammeB  of  it,  an  amount  of  heat  C  =  ea  in 
order  to  raise  its  temperature  by  1°  C.  C  is  then  called  the  atomic  heat. 

94  The  equality  of  the  mean  potential  and  the  mean  kinetic  energies  is 
true  here  as  in  the  case  of  the  linear  Planck  oscillator  (vibrating  electron), 
of.  note  41.  This  equality  is,  in  general,  always  present  when  the  forces 
which  act  upon  the  atoms  and  restore  them  to  their  positions  of  rest 
(zero  positions)  are  Ivnea/r  functions  of  the  relative  displacements  of  the 
atoms,  that  is,  when  the  force  is  "quasi-elastic,"  that  is,  proportional 
to  the  displacement  from  the  zero  position.  Cf.  in  this  connexion 
L.  Boltzmann,  Wiener  Ber.  63  (11),  731  (1871),  and  F.  Richarz,  Wied. 
Ann.  67,  702  (1899). 

WDulong  and  Petit,  Ann.  de  chim.  et  de  phys.  10,  395  (1819). 

96  The  quantity  usually  obtained  by  measurement  is  not  the  atomic 
heat  at  constant  volume  C,,,  but  the  atomic  heat  at  constant  pressure  Cf. 
For  this  we  get  values  which  in  general  fluctuate  about  the  value  6'4 
cal./deg.      The  calculation  of  Cc  from    Cp  is  based  on  the  thenno- 
dynamically  deduced  formula 

Cp  -  C,  =  a*VT 

where  a  is  the  cubical  coefficient  of  thermal  expansion,  K  the  (isothermal) 

cubical  compressibility,  and  V  the  atomic  volume  =  atomic  weight. 

density 

97  E.g.  we  find 

for  silver  at    0°  C Cp  =  6-00 

„   aluminium   „  58°  0 Cp  =  5-82 

„   copper          „   17°  C Cp  =  5-79 

„   lead  „  17°  0 Cp  =  6'33 

„   iodine  ,,   25°  C Cp  =  6-64 

„   zinc  „   17°  C Cp  =  6-03 

98  F.  E.  Weber,  Poggend.  Ann.  147,  311  (1872) ;  154,  367,  553  (1875). 

99  As  a  possible  way  out,  the  "agglomeration  hypothesis,"  supported  by 
F.  Richarz  [Marburger  Ber.  1904,  p.  1],  C.  Benedicks  [Ann.  d.  Phys.  42, 
183  (1913)]  and  others,  has  been  put  forward.     According  to  this,  as  the 
temperature  falls  the  number  of   degrees   of  freedom   of   the  system 
diminishes  by  "  freezing-in,"  as  it  were,  in  that  certain  linkages  become 
completely  rigid.     According  to  this,  however,  the  compressibility  should 
decrease  greatly  as  the  temperature  falls,  which,  according  to  E.  Orii- 
neisen's  measurements  is  not  the  case  [Verb.  d.  deutsch.  phys.  Ges.  13, 
491  (1911)].      Compare  also  in  this  connexion  the  report  of  E.  Schro- 
dinger  quoted  in  note  92. 

100.4.  Einstein,  Ann.  d.  Phys.  22,  180,  800  (1907). 

101  Cf.  A.  Einstein,  Ann.  d.  Phys.  35,  683  ff.  (1911),  also  the  report  by 
Einstein  at  the  Solvay  Congress  in  Brussels,  1911 ;  see  A.  Eucken,  Die 
Theorie  der  Strahlung  und  der  Quanten.  Abhandl.  d.  deutsch.  Bunsen-Ges., 
Nr.  7  (W.  Knapp,  Halle  1914),  pp.  330  et  seq. 

1024.  Einstein,  Ann.  d.  Phys.  34,  170,  590  (1911) ;  35,  679  (1911). 

IWThe  nature  of  the  dependence  of  the  frequency  v  on  the  three 


NOTES  AND  REFERENCES  147 

quantities  A,  p,  K  may,  according  to  Einstein  (loc.  cit.),  be  obtained  by  a 
simple  dimensional  calculation.  If  we  assume  that  v  depends  only  on  the 
mass  in  of  the  atoms,  their  distance  apart  d,  and  the  compressibility  K  of 
the  body,  then  an  equation  of  the  following  form  must  hold 

v  =  C  .  mx  .  &  .  Kz. 

C  is  here  a  numerical  constant  ;  x,  y  and  z  are  numbers  which  remain  to 
be  determined. 

The  dimensions  of  the  frequency  \y]  are  \t  -  1]  ;  the  dimensions  of  m  and 
d  are  \ni]  and  [i],  and  the  dimensions  of  the  compressibility  K  follow  from 
its  definition  : 

_  _  increase  in  volume  _ 
increase  in  pressure  x  original  volume 

K  has  therefore  the  dimensions 

I!  =  [-surface  "I 
->"-"  '      L  force  J      L 

We  thus  get  the  following  dimensional  equation 

t~l 
Hence 

x  -  2  =  0;        y  -f  z  =  0  ;        "2z  =  -  1 
from  which  we  get 

x-  -J;       y-  +4;       2-  -4 

We  have  therefore, 


Let  N  be  Avogadro's  number,  i.e.  the  number  of  atoms  in  the  gramme- 
atom.  Then  the  atomic  weight  of  the  body  is  numerically  equal  to  the 
mass  of  the  gramme-atom,  i.e. 

A  =  mN. 

If  we  imagine  the  atoms  arranged  upon  a  cubical  space-lattice  with 
sides  d,  then  the  density  must  satisfy  the  equation 


from  this  it  follows  that 

and  hence 

i. 

from  which,  it  follows  that 


148  THE  QUANTUM  THEORY 

Einstein  determines  the  factor  C  by  assuming  simply  that  only  the 
twenty-six  neighbouring  atoms  act  upon  the  displaced  atom. 

IMF.  A.  Lindemann,  Phys.  Zeitschr.  11,  609  (1910).  Lindemann's 
formula  may  be  shortly  deduced  thus  :  Let  r  =  a  sin  (2-irvt)  be  the  elonga- 
tion of  an  atom  which  is  vibrating  with  the  amplitude  a  and  the  frequency 
y.  The  mean  energy  of  this  atom  is 

E  '  f  (ar)" +  f ' 

At  the  melting-point,  according  to  Lindemann's  conception,  a  is  of  the 
same  order  as  d  (distance  apart  of  atoms).  On  the  other  hand,  the  mean 
energy  of  the  atoms  at  high  temperatures  =  'SkT,  or,  at  the  melting-point 
3kTs.  (The  melting-point,  as  a  rule,  is  high.)  From  this  it  follows  that 

=  SkT, 


But  we  have  (see  note  103) 

m  =^j-        d  =  m*,-i  =  AlN-*f-*> 
Hence 

v  =  const.  I7,*  .  4~*JSM-*»V  =  const.  T,* .  4~*  .  />*. 

105  E.  Qrilneisen,  Ann.  d.  Phys.  39,  291  et  seq.  (1912). 

106  E.  Madelung,  Nachr.  d.  kgl.  Ges.  d.  Wiss.  zu  Qottingen,  mathem.- 
physikal.  Klasse  1909,  p.  100,  and  1910,  p.  1. 

107  W.  Sutherland,  Phil.  Mag.  (6),  20,  657  (1910). 

108  If  n  and  K  are  the  coefficients  of  refraction  and  extinction  of  a 
substance  respectively,  then,  according  to  Maxwell's  Theory,  its  reflect- 
ing power  is 

n  - 


If  we  require  the  point  of  maximum  reflection,  we  have  to  form  the 
equation  ^^  =  0,  which  gives  after  reduction  the  following  relation : 

From  this  we  see  that  the  position  of  maximum  reflection  does  not 

coincide  exactly  with  the  position  of  maximum  absorption  (^  =  Oj, 

\ov        / 

but  that  it  lies  the  nearer  to  it,  the  less  the  coefficient  of  refraction 
varies  with  the  frequency.  On  the  other  hand,  the  point  of  maximum 
-absorption  lies,  according  to  the  dispersion  theory,  in  the  immediate 
^neighbourhood  of  the  natural  frequency  v,.. 

109  H.  Rvtens  and  E.  F.  Nichuh,  Wiedem.  Ann.  60,  418  (1897).     Also 


NOTES  AND  REFERENCES 


149 


77.  Rubens  and  H.  Hollnagel,  Ber.  d.  kgl.  preuss.  Akad.  d.  Wiss.  1910, 
p.  45;  //.  Hollnagel,  Dissert.  Berlin  1910;  H.  Rubens,  Ber.  d.  kgl. 
preuss.  Akad.  d.  Wiss.  1913,  p.  513 ;  H.  Rubens  and  H.  v.  Wartenberg, 
ibid.,  1914,  p.  169. 

As  an  example  we  give  here  the  following  small  table  in  which  \ 
denotes  the  wave-length  of  the  "  residual  "  rays,  as  given  by  the  above 
investigators. 


A. 

A. 

NaCl 
KC1 
AgCl 
HgCl 

52   M 

63-4M 
81  -5M 
98-8/i 

T1C1 
KBr 
AgBr 
TIBr 

91-6M 
82-6M 
112-7M 
117   M 

HO  Of.,  however,  note  108. 

111  W.  Nernst  and  F.  A.   Lindemann,   Sitzungsber.  d.  kgl.  preuss. 
Akad.  d.  Wiss.  1911,  p.  494  ;  W.  Nernst,  Ann.  d.  Phys.  36,  426  (1911). 

112  The  following  short  table  gives  the  values  for  v  which  are  calculated 
from  Einstein's   formula  (35),   Lindemann's  formula  (36),  from   the 
"residual  rays"  (see  note  109),  and  from  the  observed  atomic  heat 
according    to    an    empirical    formula    (40)    proposed  by  Nernst    and 
Lindemann.     For  more  detailed  data  with,  in  part,  corrected  numerical 
factors  see  C.  E.  Blom,  Ann.  d.  Phys.  42,  1397  (1913). 


Substance 

VE 

"L 

''residual  rays 

''atomic  heat 
(Nerntt-Lindemann) 

Al 

6-7  .  1012 

7-6  .  10" 

8-3  .  10" 

Cu 

5-7  .  1012 

6-8  .  1012 

6-7  .  1012 

Zn 

4-4  .  1012 

4-8  .  1012 

Ag 

4-1  .  1012 

4-4  .  1012 

4-5.1012 

Pb 

2-2  .  1012 

1-8.  1012 

1-5  .  1012 

Diamond 

32-5  .  1012 

40  .  1012 

NaCl 

7-2.  1012 

5-8  .  1012 

5-9  .  1012 

KC1 

5-6  .  1012 

4-7  .  1012 

4-5  .  1012 

113  W.  Nernst,  F.  Koref,  F.  A.  Lindemann,  Untersuchungen  iiber  die 
spezifische  Warme  bei  tiefen  Temperaturen.  I.  u.  II.  Sitzungsber.  d. 
kgl.  preuss.  Akad.  d.  Wiss.  1910,  3  March.— W.  Nernst,  idem  III.,  ibid., 
1911,  9  March. — F.  A.  Lindemann,  idem  IV.,  ibid.,  1911,  9  March.— W. 
Nernst  and  F.  A.  Lindemann,  idem  V.,  ibid.,  1911,  27  April.— W.  Nernst 
and  F.  A.  Lindemann,  idem  VI.,  ibid.,  1912,  12  Dec.— W.  Nernst,  idem 
Vn.,  ibid.,  1912,  12  Dec.— W.  Nernst  and  F.  Schwers,  idem  VIII.,  loc. 
tit.,  1914.— W.  Nernst,  Der  Energieinhalt  fester  Stoffe.  Ann.  d.  Phys. 
36,  395  (1911). 

HI  W.  Nernst,  Die  theoretischen  und  experimentellen  Grundlagen  des 
neuen  Warmesatzes.  (W.  Knapp,  Halle  1918.) 

115  The  First  Law  states  :  If  d'Q  is  the  heat  supplied  to  a  system,  d'A 


150  THE  QUANTUM  THEORY 

the  work  done  on  the  system  from  outside,  then  the  increase  of  energy 
U  of  the  system  is  given  by 

dU  =  d'Q  +  d'A. 

The  Second  Law  states  :  if  d'Q  is  supplied  reversibly  at  the  temperature 
T,  then  —^  is  the  complete  differential  of  the  entropy  S,  hence 


Let  us  follow  Helmholtz  and  introduce  the  "  free  energy"  F  denned  by 

F=  U-  T-  S. 
Then  it  follows  that 

dF  =  dU  -  T.dS  -  S.dT=d'Q  +  d'A  -  T.dS  -  S.dT 

i.e.  dF  =  d'A  -  S  .  dT 

for  every  reversible  process. 

If  the  process  is  isothermal  (dT  =  0)  then  it  follows  that  dF  =  d'A 
or,  for  a  finite  change  of  state,  F.A  -  Fl  =  A.  If  we  set  A'  =  -  A,  so 
that  A'  is  the  work  gained,  we  get 

Fj  -  F,  =  A'. 

That  is,  the  work  gained  in  the  isothermal  reversible  process  —  which  is, 
as  may  be  shown,  the  maximum  obtainable  —  is  equal  to  the  decrease  of 
free  energy. 

Further,  it  follows,  since  at  constant  volume  F  the  work  d'A  =  0, 
that 


Therefore,  formulating  these  expressions  for  two  states,  we  get 
/pjOffii  -  "a)~|    _  ^  _  p^  _  ^J7}  _  {/2j 

or,  finally,  if  we  write  for  short  t7j  -  Uz  =  U' 

an  equation  much  used  in  physical  chemistry. 
Since,  now,  according  to  Nernst's  heat  theorem, 

(§'),.         =o 

\OT  Jhm  T=Q 
(A'  -  U')  vanishes  for  T  =  0,  being  above  the  first  order. 

Hence  lim  ?L4' ~ .  ^')  =  Q 

and  hence  also 


NOTES  AND  REFERENCES  151 

This  is  equation  (89)  of  the  text. 

From  §£  -=  -  S,  it  follows  further  that  ^F] '  "  F^  -  S2  -  8lt  or 

Q     _    O     _  9-4' 

and  hence  Nernst's  Theorem  may  be  formulated  thus 

lim  (S2  -  S,)  =  0 

r=o 

that  is,  in  £/«;  neighbourhood  of  the  absolute  zero  all  processes  proceed 
without  change  of  entropy. 

116  Cf.,  for  example,  M.  Planck,  Lectures  on  Thermodynamics.  Planck 
goes  further  than  Nernst  inasmuch  as  he  postulates  that  not  only  the 
difference  of  the  entropies  S2  -  Sx  is  zero  at  absolute  zero  (see  previous 
note)  but  also  that  the  individual  values  themselves  become  zero.  Hence, 
according  to  Planck,  at  the  absolute  zero  of  temperature  the  entropy  of 
every  chemically  homogeneous  body  is  equal  to  zero.  From  this  the  con- 
clusion given  in  the  text, 

lim^^,j  =  0 

may  be  deduced  immediately.     It  follows  from  the  relation  (occurring 
in  the  last  note) 

F  -  U  =  -  TS 

and  from  Planck's  version  of  Nernst's  Theorem,  that  F  -  0"  vanishes  for 

T  =  0,  being  of  higher  order  than  the  first. 

Hence 


=  0     or     limf^-H  S) 
T=o\oT        ) 


or,  finally, 


117  For  low  temperatures,  that  is,  for  high  values  of  x  =  ^  Einstein's 

formula  (34)  takes  the  following  form  :  Cv  =  3Rx*e~x.  The  falling-off  at 
low  temperatures  therefore  follows  an  exponential  law ;  more  exactly,  it 
varies  as 

i        const 

118  W.  Nernst  and  F.  A.  Lindemann,  Sitzungsber.  d.  kgl.  preuss.  Akad. 
d.  Wiss.  1911,  p.  494 ;  Zeitschr.  f.  Elektrochemie,  17,  817  (1911). 

119  A.  Einstein,  Ann.  d.  Phys.  35,  679  (1911). 

120  For  if  we  regard  the  atoms  as  mass-points,  then  each  atom  hao  three 
degrees  of  freedom ;  the  whole  body  has  therefore  '3N  degrees  of  freedom. 
As  is  proved  in  mechanics,   however  (cf.   R.  H.   Weber  and  R.  Oans, 


152  THE  QUANTUM  THEORY 

Repertoriuin  der  Physik  Bd.  I.  pp.  175  et  seq.),  a  mechanical  system  of 
3N  degrees  of  freedom  has  3N  natural  frequencies,  and  the  moit  general 
small  motion  of  each  atom  consists  in  a  superposition  of  these  3N  natural 
frequencies. 

121  P.  Debye,  Ann.  d.  Phys.  39,  789  (1912). 

122  M.  Born  and  Th.  v.  Kdrmdn,  Phys.  Zeitschr.  13,  297  (1912);  14, 
15,  65  (1913).     Of.  also  M.  Born,  Ann.  d.  Phys.  44,  605  (1914) ;  M.  Born, 
Dynamik  der  Kristallgitter  (Teubner,  Leipzig  and  Berlin  1915). 

123  Of.,  for  example,  R.  Ortvay,  tjber  die  Abzahlung  der  Eigenschwin- 
gungen  fester  Korper.     Ann.  d.  Phys.  42,  745  (1913). 

Ortvay  considers  the  natural  frequencies  of  an  elastic  cube,  each  side 
of  which  has  the  length  L.  There  are  found  to  be  three  groups  of  natural 
frequencies.  The  first  two  groups  are  the  transversal  frequencies,  the 
third  group  is  the  group  of  the  longitudinal  frequencies.  That  the  trans- 
versal frequencies  form  two  groups  (moreover  identical)  is  easily  seen. 
For  in  the  case  of  a  transversal  vibration,  which  is  propagated  in,  say,  the 
direction  of  the  aj-axis,  two  equal  alternatives  are  probable,  namely,  that 
the  particles  vibrate  parallel  to  the  y-  or  to  the  2-axis.  In  the  case  of  the 
longitudinal  oscillations,  however,  there  is  naturally  only  one  group ;  for 
in  the  case  of  propagation  along  the  z-axis  there  is  only  one  possibility, 
namely,  that  the  particles  vibrate  parallel  to  the  x-axis.  The  frequencies 
of  the  first  two  groups  are  characterised  by  the  values 


the  third  group  by 


Here  ct  and  ct  are  the  velocities  of  propagation  of  transversal  and  longitu- 
dinal waves  in  the  body,  whereas  a,  b,  c  are  arbitrary  positive  whole 
numbers.  If  therefore  we  give  a,  b,  c  all  possible  values  in  all  possible  com- 
binations, we  get  all  the  possible  transversal  and  longitudinal  natural  fre- 
quencies, which  together  form  the  elastic  spectrum  of  the  cube.  If  now 
we  inquire  how  many  transversal  natural  frequencies  of  the  first  group 
fall  below  v,  this  means  nothing  else  than  inquiring  how  many  trios  of 
values  (a,  b,  c)  fulfil  the  condition 


Imagine  a,  b,  c  as  co-ordinates  of  a  point  in  space.  Then  all  possible  trios 
(a,  b,  c)  of  values  are  represented  by  the  total  "lattice-points"  of  the 
positive  space  octant,  and  the  above  question  is  answered  by  counting  how 

many  lattice-points  are  at  a  distance  less  than  —  ''from  the  origin  (0,  0,0). 

c 


NOTES  AND  REFERENCES  158 

All  these  lattice-points  lie  within  the  positive  octant  of  the  sphere  whose 

radius  is  —  -.     Since  now  one  lattice-point  is  assigned  to  every  volume  of 

ct 

magnitude  1  —  namely,  every  elementary  cube  —  the  required  number  of 
lattice-points,  provided  that  it  is  sufficiently  large,  is  equal  to  the  volume 

of  the  positive  spherical  octant  of  radius  —  ",  i.e.  is  equal  to 


If  V  =  L3  is  the  volume  of  the  given  cubical  body,  then  the  number 
of  the  transverse  natural  frequencies  below  v  belonging  to  the  first 
group  is 

Zl  =  T  FcJ 

The  number  belonging  to  the  second  group  is  the  same,  that  is 

rr  r?          4ir  Tr  V 


Finally,  the  number  of  the  longitudinal  frequencies  corresponding  to 
these  is 

4ir      v"' 


We  thus  get  for  the  total  of  all  natural  frequencies  below  v 


The  total  of  natural  frequencies  in  the  interval  v  .  .  .  v  +  dv  follows  by 
differentiation  with  respect  to  v 


and  this  is  just  formula  (43)  of  the  text. 

124  In  formula  (43)  for  Z(v)dv  let  us  replace,  according  to  formula  (44) 
of  the  text,  the  factor 


Then  it  follows  that 


154  THE  QUANTUM  THEORY 

If  we  now  aet         -  «  and         -  «m,  we  get 


128  A  table  showing  how  the  Debye  function  C»  depends  on  xm  is  given 
by  Nernst  (Die  theoretischen  und  experimentellen  Grundlagen  des  neuen 
Warmesatzes.  W.  Knapp,  Halle  1918,  p.  201).  In  it  the  simple 
Einstein  function  [formula  (34)  of  the  text]  is  also  tabulated. 

126  If  T  is  great,  then  xm  is  small  compared  with  1  ;  then  we  may 
replace  in  the  integral  of  (45)  ex  by  1  in  the  numerator,  and  ex  -  1  by  x 
in  the  denominator.  It  then  follows  that 


127  If  Tia  small,  then  xm  is  large,  and  we  may  replace  the  upper 
limit  of  the  integral  as  a  first  approximation  by  oo  .  The  integral  will 
thus  become  a  numerical  constant  independent  of  xm,  and  it  follows  that 


C.  -  j*  .  const.  -9-«  .  T3  .  const. 
128  From  the  theory  of  elasticity  it  follows  that 
and         c  = 


J  HE 
\H-o 


where  K  is  the  compressibility,  p  the  density,  and  a  the  ratio 

transverse  contraction 
longitudinal  dilatation* 

If  we  insert  these  values  in  (44)  and  note  further  that  V  =  -,   formula 

(46)  of  the  text  follows. 

129  As  the  number  of  frequencies  below  v  is  proportional  to  v*,  we  get, 
for  example,  the  following  picture  :  if  we  divide  the  interval  from  0  to 
vm  into  10  parts,  and  if  only  one  natural  frequency  lies  in  the  first 
division,  then  in  the  following  divisions  there  will  be  7,  19,  37,  61,  91, 
127,  169,  217,  271  natural  frequencies  ;  i.e.  the  natural  frequencies  crowd 
continually  closer  together. 

180  P.  Debye,  Ann.  d.  Phys.  39,  789  (1912)  ;  W.  Nernst  and  F.  A. 
Lind&mann,  Sitzungsber.  d.  Berl.  Akad.  d.  Wiss.  1912,  p.  1160. 

131  A.  Eucken  and  F.  Schwers,  Verhandl.  d.  deutsch.  physikal.  Ges. 
15,  578  (1913)  ;  W.  Nernst  and  F.  Schwers,  Sitzungsber.  d.  Berl.  Akad.  d. 
Wiss.  1914,  p.  355  ;  P.  Grttnther,  Ann.  d.  Phys.  51,  828  (1916)  ;  W.  H. 


NOTES  AND  REFERENCES  155 

Keesom  and  Kmnerlingh-Onnes,  Amsterdam  Proc.  17,  894  (1915).     Cf. 
also  the  graphic  tables  by  E.  SchrOdinger,  Phys.  Zeitschr.  20,  498  (1919). 
182  If  we  introduce  into  equation  (44)  of  the  text, 


a  "  mean  acoustic  velocity  "  c,  by  the  obvious  definition 


then  for  the  order  of  magnitude  of  the  smallest  wave-length  \min,  there 
follows 

c 


It  now  the  atoms  in  the  cubical  space-lattice,  for  example,  are  arranged 
so  as  to  be  a  distance  a  apart,  then  Na*  =  V,  and  hence 


•VI- 


Amin 

\  3 

133  For  references  see  note  122. 
131  Of.  Born,  Dynamik  der  Kristallgitter,  §  19. 
133  .F.  Haber,  Verb.  d.  deutsch.  phys.  Ges.  13,  1117  (1911). 
For  if  the  atomic  residue  (mass  m)  and  the  electron  (mass  /i)  are  held 
to  their  zero  positions  by  forces  of  the  same  order  of  magnitude,  and  if 
they  vibrate  independently  of  one  another  (a  simplifying  supposition)  the 
equation  of  vibration  of  the  atom  is  nix  +  a?x  =  0,  the  solution  of  which  is 

x  =  A  sin(  ~~F^t  ).     The  infra-red  frequency  of  the  atom  is,  therefore, 

a 
yr=2a-   /—  ,  and  correspondingly,  the  ultra-violet  frequency  of  the  electron 

Hence  Habeas  Law  follows  :    vr  :  vv  =  <s//x  :  N/»».      The 


general  space-lattice  theory  of  M.  Born  confirms  this  law  and  shows  that 
in  the  lattice,  too,  atomic  residues  and  electrons  appear  upon  an  equal 
footing,  and  are  acted  upon  by  forces  of  the  same  order  of  magnitude. 

136  Cf.  M.  Born  and  Thos.  v.  Kdrmdn,  Phys.  Zeitschr.  13,  297  (1912). 

We  may  treat  this  problem,  which  is  of  course  one-dimensional,  most 
simply  thus  :  If  we  imagine  an  endless  chain  of  points  of  equal  mass  TO 
disposed  along  the  x-axis  at  a  distance  apart  a,  and  if  we  suppose  for 
simplicity  that  each  mass-point  only  acts  upon  its  two  neighbours,  then 
the  equation  of  motion  of  the  nth  point  is 

mxn  =  a(xn+i  -  Xn)  -  a(xlt  -  SC,,_i)  =  a(xn+i  +  X«_i  -  2«n). 
Here  o  is  a  constant,  and  n  can  assume  all  values  between  +  o>and  -  <n  . 
As  a  solution  let  us  set  for  trial 

xn  =  A  sin    2*yt  -  n\ 


156  THE  QUANTUM  THEORY 

This  represents  a  process  which  is  periodic  in  space  and  time,  that  is,  a 
wave  which  is  propagated  along  the  chain  in  the  direction  of  increasing 
x.  The  frequency  of  this  wave  is  v,  its  length  is  A.  Then,  if  after  p 
points  the  same  displacement  is  to  recur,  pa  must  =  A,  and  hence  it 
actually  follows  that 

f  2*-a" 

xn+J>  =  A  sin     2irvt  -  (n  +  p)  — 


In  order  to  find  the  relation  between  v  and  \  (that  is,  the  "  law  of  dis- 
persion"), let  us  insert  the  above  formula  in  the  equation  of  motion. 
Then  it  follows  that 


-  (n  +  1)—  1  +  sin  [znt  -  (n  -  1)—  1 

*  J         L  A  J 

-n^l\ 
*  Ji 


-  2*1  sin 


-  £*]  .  (l  -  cos  *2\ 
\  J        \  A    / 


That  is, 


„  =  *  J«~sin  («)-  ,m  sin  (««Y  if  we  set  LJt  =  ,m. 
W  \w       \A/  \A/  ir  Vro 

187  Of.  .Bom,  Dynamik  der  Kristallgitter,  p.  51. 

From  the  special  case  treated  in  the  previous  note,  we  also  recognise 
the  truth  of  law  (49)  ;  for  if  A  is  much  greater  than  a,  the  dispersion  law 

takes  the  form  v  =  n»—  «••»,  where  q  =  vmita,  represents  the  velocity  of 

A  A 

propagation  of  the  wave,  and  this  is  independent  of  the  wave-length. 

138  The  statement  that  a  given  direction  lies  in  the  element  of  solid 
angle  dti  is  intended  to  convey  the  following  sense  :  about  an  arbitrary 
origin  O  describe  a  "  unit  sphere,"  i.e.  a  sphere  of  radius  1.    Now  let  a  cone 
of  infinitely  small  angle  be  constructed  of  rays  passing  through  0,  the  point 
of  the  cone  lying  at  O.    Let  this  cone  cut  out  of  the  surface  of  the  unit 
sphere  a  small  element  of  surface  dn.    Now  let  the  parallel  ray  to  the 
"given  direction"  be  drawn  through  O  (here,  for  example,  the  wave- 
normals).     If  this  ray  lies  in  the  cone  just  constructed,  then  we  say  that 
the  "  given  direction  "  lies  in  the  elementary  solid  angle  dfl. 

139  The  capacity  for  heat  of  a  certain  finite  body  is  that  amount  of  heat 
which  must  be  imparted  to  the  whole  body  in  order  that  its  temperature 
be  raised  by  1°  C.     If  M  is  the  mass  of  the  body,  and  c  its  specific  heat, 
then  its  capacity  for  heat  is 


NOTES  AND  REFERENCES  157 

From  the  mean  energy  content  E  of  the  whole  body,  r  follows  by 
differentiation  with  respect  to  the  temperature 

r-*1 

140  This  somewhat  complicated  calculation  runs  as  follows :  we  start 
from  the  formula 


and  first  replace  \  by  g.     Thus  we  get 


and  the  integral  with  respect  to  A.  is  transformed  into  one  with  respect  to 
vi.    The  limits  of  this  integral  are 

vi  =  g*'(n)         [corresponding  to  \  =  A^n)] 

and 

vi  =  0         (corresponding  to  A.  =  oo ). 
If  we  further  set 

we  get 

3      47T  *»<»> 

fe'T3^  C  dn    ;'x4e*dx 


In  place  of  the  quantities  gi(»i)  and  x»»(n)  which  still  depend  essenti- 
ally on  the  direction,  let  certain  mean  values  be  introduced.     Firstly,  let 

us  set 

4ir 


In  this  way  three  mean  acoustic  velocities  glf  22,  g3,  independent  of  the 
direction,  are  defmed.  We  further  introduce  in  place  of  Am(n)  a  mean 
value  independent  of  the  direction,  in  the  following  manner.  In  deduc- 
ing formula  (55)  we  saw  that 


. 

0        Am(n) 


158  THE  QUANTUM  THEORY 

If  we  carry  out  the  integration  with  respect  to  x,  we  get 

to 

v  r  da    x£f 

o 


u, 

47T 


dn 


Now,  in  a  way  analogous  to  that  used  for  the  acoustic  velocities  g,  we 
set 


Hence 


Into  ^n)  =  \  we  introduce  in  place  of  qi(n)  and  \m(n)  the  mean 


values  qi  and  \m,  which  are  independent  of  direction  ;  thereby  xi(n)  also 
becomes  independent  of  direction,  and  is  transformed  into 


It  follows  that 


'-1*1  ""' 

141  At  the  lowest  temperatures 

. 


32^2  Ji-  -  }£ 
i=iXl     b 

3  a 


NOTES  AND  REFERENCES  159 

Now  the  value  of  the  integrals  =  ^w4.     If  we  further  set  R  =  Nk,  and 
for  z~J  the  value  (59),  we  get 


15**       '  Zfe  • 

l 

If  we  introduce_in  place  of  the  three  acoustic  velocities^",  2j,  3s  a  mean 
acoustic  velocity  q  by  means  of  the  definition 


it  follows  that 


Finally  for  —  we  can  write  V^  (mean  atomic  volume)  and  thus  get  the 
formula 


148  H.  Thirring,  Phys.  Zeitsohr.  U,  867  (1913)  ;  15,  127,  180  (1914). 

143  M.  Born  and  Th.  v.  Kdrmdn,  Phys.  Zeitschr.  14,  15  (1913). 

144  Cf.  note  182. 
148  Cf.  note  128. 

1484.  Eucken,  Verhandl.  d.  deutsch.  physikal.  Ges.  15,  571  (1913).  Cf. 
also  A.  E'ucken,  Die  Theorie  der  Strahlung  und  der  Quanten  (W.  Knapp, 
Halle  1914),  pp.  386  et  seq.,  Appendix. 

147  Cf.  A.  Eucken,  Die  Theorie  der  Strahlung  und  der  Quanten  (W. 
Knapp,  Halle  1914),  p.  387. 

148  To  calculate  the  mean  acoustic  velocity  q,  the  relation  given  in  note 
141  is  used 

3  4ir 


We  have  therefore  to  obtain  from  the  "dispersion  equation"  of  the 
crystal  in  question  (for  long  waves)  the  values  of  the  three  acoustic 
velocities  q^n),  q2(n),  qs(n)  as  functions  of  the  wave-direction  ;  q  is  then 
obtained  from  the  above  formula  by  integration  over  all  directions  and 
finally  summation. 

149  L.  Hopf  and  G.  Lechner,  Verhandl.  d.  deutsch.  physikal.  Gee.  16, 
643  (1914). 

180  The  following  short  table  is  taken  from  the  paper  of  Hopf  and 
Lechner  cited  in  note  149  : 


160 


THE  QUANTUM  THEORY 


Crystal 

2  calc.  from  Cy 

5  calc.  from  elastic 
data 

Sylvin  .        .        . 
Rock  salt     . 
Fluor-spar   . 
Pyrites 

2-36  .  10B 
2-82  .  108 
4-02  .  108 
5-43  .  105 

2-03  .  10s 
2-72  .  105 
3-82  .  108 
5-12  .  105 

181  W.  Nernst,  Vortrage  iiber  die  kinetische  Theorie  der  Materia  und  der 
Elektrizitat.     Wolfskehl-Kongress  1913  in  Gottingen  (Teubner,  Leipzig 
and  Berlin  1914),  pp.  63  et  seq. 

182  W.  Nernst,  ibid.,  pp.  81  et  seg_. 

153  E.  Schrodinger,  Phys.  Zeitschr.  20,  503  (1919).  Schrodinger  correctly 
points  out  that — apart  from  the  substitution  of  one  single  mean  x  for 
the  three  quantities  x>  in  the  Debye  terms — the  approximation  above  all  in 
the  second  part  of  Cv  (i.e.  the  replacement  of  the  3(s  -  1)  frequencies 
vt  .  .  .  v3  by  the  constants  ^  .  .  .  i>§,)  may  not  be  permissible  in  many 
cases :  namely,  in  those  cases  in  which  the  masses  of  the  various  kinds 
of  atoms  are  not  very  different  from  one  another.  If  we  were  to  allow 
— so  he  argues — the  masses  of  the  different  kinds  of  atoms  and  the  forces 
acting  upon  them  gradually  to  become  equal  to  one  another,  a  simple 
atomic  lattice  would  result,  and  during  this  process  the  3(s  -  1)  branches 
of  the  spectrum,  which  correspond  to  the  second  type  of  motion,  would 
merge  into  the  three  first  branches.  "  They  cannot  therefore  even  be 
approximately  monochromatic  if  the  masses  differ  only  slightly." 

184  H.  Thirring,  Phys.  Zeitschr.  15,  127,  180  (1914). 

188  M.  Born,  Ann.  d.  Phys.  44,  605  (1914). 

186  E.  Oriineisen,  Ann.  d.  Phys.  39,  257  (1912). 

187  S.  Ratnowski,  Verhandl.  d.  deutsch.  physikal.  Ges.  15, 75  (1913). 

188  Vortrage  iiber  die  kinetische  Theorie  der  Materie  und  der  Elek- 
trizitat.    Wolfskehl-Kongresz  zu  Gottingen,  1913.     (Teubner,  Leipzig 
and  Berlin  1914),  Vortrag  P.  Debye. 

189  If  U  is  the  energy,  and  S  the  entropy  of  the  system,  then  the  "  free 
energy  "  is  defined  according  to  Helmholtz  by  the  relation 

F  =  U  -  S  .  T. 
It  then  follows  from  note  115  that 

dF  =  d'A  -  S .  dT 
where  d'A  is  the  work  done  from  without.    If  we  set  in  the  usual  way 

d'A  =  -  pdV         (p  =  pressure,  V  =  volume) 
then 

dF=-  pdV-  SdT. 

From  this  we  get  immediately  the  equation  (66)  in  the  text 


NOTES  AND  REFERENCES  161 

Similarly, 

fdF\   _  _  s 

and  hence 

160  P.  Debye,  loc.  cit.,  note  158. 

161  E.  Oriineisen,  Ann.  d.  Phys.  26,  211  (1908) ;  33,  65  (1910) ;  39,  285 
(1912). 

162 P.  Debye,  loc.  cit.,  note  158. 

1634.  Eucken,  Ann.  d.  Phys.  34,  185  (1911) ;  Verhandl.  d.  deutsch. 
physikal.  Ges.  13,  829  (1911). 

IMP.  Drude,  Ann.  d.  Phys.  1,  566  (1900). 

163  #.  Riecke,  Wiedem.  Ann.  66,  353,  545  (1898). 

166  Of.,  for  example,  H.  A.  Lorentz,  The  Theory  of  Electrons  (Teubner, 
•  Leipzig,  and  Berlin  1909). 

167  Let  g  be  the  average  velocity  of  the  electrons  along  the  free  path  I. 

Then  the  electron  takes  the  time  T  =  -  to  pass  over  this  free  path. 

During  this  time  it  is  exposed  to  the  electrical  force  E  of  the  external 
field.  Its  increase  in  velocity  due  to  this  force  is  at  the  commencement 

of  the  free  path  =  0,  at  the  end  of  it  =  ?=—,  where  e  and  m  are  the 

fli 

charge  and  mass  of  the  electron  respectively.  In  the  mean,  therefore, 
the  small  additional  velocity  generated  by  the  field  is  Ag  =  -^^-  =  |^-- 

The  electrons  stream  unidirectionally  with  this  velocity  against  the  field. 
If  N  is  the  number  of  electrons  per  unit  volume,  then  through  unit 
area  of  the  surface  there  streams  per  second  a  quantity  of  electricity 

This  is,  however,  the  "current  density"  I  which  is 


known  to  be  connected  with  the  field  E  by  the  relation  I  =  ffE-     The 
expression  (67)  for  the  conductivity  <r  therefore  follows. 

A  more  thorough  treatment  is  due  to  H.  A.  Lorentz  (see  note  166). 
He  does  not  give  the  electrons  a  single  velocity  q,  but  introduces  Max- 
well's supposition,  known  from  the  kinetic  theory  of  gases,  that  all 
possible  velocities  occur,  which  are  distributed  among  the  electrons 
according  to  a  fixed  law,  the  so-called  Maxwell  Law  of  Distribution. 
He  thus  obtained  a  formula  of  the  following  form  : 


•- 

\3ir   mq 

which  therefore  only  differs  by  a  numerical  factor  from  Dnide's  formula 
(67)  ;  here  q  =   v  2a,  the  root  mean  square  of  the  velocity. 

168  Let  a  temperature  gradient  along  the  x  axis  be  present  in  the  piece 
of  metal.      Let  a  section  be  taken  (see  Pig.  12)  at  right  angles  to  the 
11 


162  THE  QUANTUM  THEORY 

x  axis;  we  shall  calculate  the  energy  transport  across  this  section  per 
second.  If  we  suppose  that  £  of  all  electrons  wander  in  each  of  the  three 
directions  in  space,  then  £  move  in  the  positive  x  direction ;  and  further, 
^_  the  number  of  electrons  which 

pass  through  the  unit  of  sur- 
face in  one  second,  will  be  all 
those  which  are  contained  in 
the  small  shaded  cylinder  with 
the  base  surface  area  1  and  the 
height  q  (velocity),  namely, 


FIG.  12.  iNg.     We  also  make  the  sup- 

position, usual  in  the  theory  of 

gases  (although  not  strictly  true),  that  the  energy,  which  each  electron 
transports  through  the  cross-section,  has  the  value  corresponding  to  that 
which  it  had  at  the  point  where  it  last  collided. 
Now  the  energy  in  the  section  itself  at  temperature  T  is  equal  to  %kT, 

and  hence  the  energy  =  |fcr  +  ^j^  .  I  at  the  points  which  lie  at  a 

distance  I  in  front  of  and  behind  the  section.     Here,  on  the  average,  the 
electrons  coming  from  the  right  and  the  left  meet  with  their  last  collisions. 
The  energy  transport    per    second    through  unit  of  cross-section   is 
therefore 


Hence  7  =  ^JUlqk  is  the  coefficient  of  thermal  conductivity. 

Here  also  U.  A.  Lorentz  has  deepened  the  theory  by  taking  the  distri- 
bution of  velocity  into  account,  and  finds  that 


where  again  q  =  v  g2 

169  G.  Wiedemann  and  R.  Franz,  Poggend.  Ann.  89,  497  (1853)  ;  L. 
Lorenz,  Wiedem.  Ann.  13,  422,  582  (1881).  Of.  also  G.  Kirchho/  and 
G.  Hansemann,  Wiedem.  Ann.  13,  417  (1881);  W.  Jaeger  and  H. 
Diesselhorst,  Abh.  d.  phys.  techn.  Beichsanstalt  3,  269  (1900). 

The  following  short  table  is  taken  from  the  paper  of  the  two  investi- 
gators last  named  ;  it  gives  the  ratio  "L  for  various  metals  at  a  temperature 
of  18°  C. 


NOTES  AND  REFERENCES 


163 


Metal 

y  .  10-10 

a- 

Al 

6-36 

Cu 

6-65 

Ag 

6-86 

Au 

7-09 

Zn 

6-72 

Pb 

7-15 

Pt 

7-53 

Bi 

9-64 

170  It  follows  from  (67)  by  setting  ^mq2 


,  that  is,  a  = 


—,  that 
ra 


Now,  let  JY  be  the  number  of  atoms  per  unit  volume,  .AT*  the  number 
of  atoms  in  a  gramme-atom  (Avogadro's  number).  If,  further,  A  is  the 
atomic  weight,  M  the  mass  of  an  atom,  and  p  the  density,  then 

(A  =  MN* 
\p  =  MN 
therefore 


We  next  assume  that  N>  the  number  of  electrons  per  unit  of  volume,  is 
small  compared  with  N,  say 


If  we  insert  this  value,  then  we  get  for  the  free  path 


200  <r 


We  shall  make  a  rough  calculation  for  copper  at  0°  G.     We  have 

ff  <-->  5-4 . 1017  (in  electrostatic  units) 
A  =  63-57 
k  =  1-4  . 10- 16 
T=  273 
m  =  0-9  . 10- 27 
N*  =  6-1 . 1023 
P  =  8-9 
e  =  4-77  . 10-10. 

With  these  values  we  get 

I  is  of  the  order  5-7  . 10-". 


164  THE  QUANTUM  THEORY 

Since  the  atomic  distance  is  of  the  order  of  magnitude  2 .  10~8,  the 
electrons  would  therefore  only  suffer  collision  after  passing  many  thou- 
sands of  atoms.  This  is  unacceptable,  since  the  "radius  of  molecular 
action  "  of  the  atoms  itself  has  dimensions  which  fall  within  the  order  of 
magnitude  of  about  10  ~8. 

171  JET.  A.  Lorentz,  loc.  cit.,  note  43. 

172  J.  J.  Thomson,  The  Corpuscular  Theory  of  Matter. 

173  H.  Kammerlingh-Onnes,  Leiden  Communicat.  1913,  133. 
171  C.  H.  Lees,  Phil.  Trans.  (A)  208,  381-443  (1908). 

175  W.  Meissner,  Ann.  d.  Phys.  47,  1001  (1915). 

176  W.  Nernst,  Berl.  Ber.  1911,  p.  310. 

177  H.  Kammerlingh-Onnes,  Leiden  Communicat.  119,  22  (1911). 

178  F.  A.  Lindemann,  Berl.  Ber.  1911,  p.  316. 

179  W.    Wien,  Berl.  Ber.   1913,   p.   184.      Of.   also   Vorlesungen   iiber 
neuere  Probleme  der  theoretischen  Physik.     (Teubner,  Leipzig  and  Berlin 
1913.)     3.  Vorlesung. 

180  If  s  is  the  radius  of  atomic  action,  N  the  number  of  stationary  atoms 
per  unit  of  volume,  then,  according  to  a  well-known  result  of  the  kinetic 
theory  of  gases,  the  mean  free  paths  of  the  electrons 


Let  us  set 


where  s0  is  the  radius  of  atomic  action  for  T  =  0,  that  is,  when  the  atoms 
are  at  rest ;  let  a  be  the  amplitude  of  atomic  vibration.  Now  the  mean 
energy  E  of  this  vibration  (frequency  i>),  on  the  one  hand,  =  —  (2irv)*az 

(M  is  the  atomic  mass) ;  on  the  other  hand,  it  is,  according  to  Planck- 
Einstein, 


From  this  it  follows  that 


Now,  according  to  formula  (67)  of  the  text,  the  resistance 

w    *      2m2 
r*r-I* 

If  we  here  set  for  q  the  value,*/ —  (cf.  note  170),  and  for  N,  according 

to  J.  J.  Thomson's  supposition,  a^T,  and  for  =-  the  value 
•*Ni*  =  *N(a'l  +  2as0  +  si) 


NOTES  AND  REFERENCES  165 

it  follows  that 


an  expression,  which  contains  only  a  and  s0  as  unknown  constants.    If 
we  set 


then  W  assumes  the  form  given  in  the  formula  (70). 

181  F.  A.  Lindemann,  Phil.  Mag.  29,  127  (1915). 

181a  F.  Haber,  Berl.  Akad.  Ber.  1919,  pp.  506  and  990. 

182  J.  -Stark,  Jahrb.  d.  Radioakt.  u.  Elektronik  9,  188  (1912). 

183  G.  Borelius,  Ann.  d.  Phys.  57,  278  (1918). 

184  K.  Herzfeld,  Ann.  d.  Phys.  41,  27  (1913). 

185  If  we  set  %mq*  =  E,  therefore  q  =  \K,  the  first  of   the  two  for- 


mulae  (72)  follows  from  (67).  If  we  further  take  into  account  that  in  Drude's 

9   /7  771 

Theory  E  =  f  kT,  that  is,  that  &  -  |  ^  then  from  (68)  the  second  for- 
mula (72)  follows. 

186  F.  v.  Hauer,  Ann.  d.  Phys.  51,  189  (1916). 

187  W.  Nernst,  Berl.  Ber.  1911,  p.  65. 
1884.  Eucken,  Berl.  Ber.  1912,  p.  141. 

189  K.  Scheel  and  W.  Heuse,  Ann.  d.  Phys.  40,  473  (1913).  Of.  also 
L.  Holborn,  K.  Scheel  and  F.  Henning,  Warmetabellen  der  physikal.- 
techn.  Reichsanstalt  (Vieweg  1919). 

1904.  Einstein  aud  O.  Stem,  Ann,  d.  Phys.  40,  551  (1918). 

191  The  quantum  formulae  (76)  and  (77)  properly  correspond  to  the 
Planck  oscillator,  that  is,  to  a  system  of  one  degree  of  freedom,  while 
here,  in.  the  case  of  rotation,  we  have  to  do  with  two  degrees  of  freedom. 
But  the  energy  of  the  Planck  oscillator  is  composed  of  two  equal  parts,  a 
kinetic  and  a  potential  part,  while  in  the  case  of  rotation  only  kinetic 
energy  comes  into  question.     This  is  often  expressed  thus  :  the  Planck 
oscillator  possesses  one  potential  and  one  kinetic  degree  of  freedom,  while 
the  rotating  molecule  possesses  two  kinetic  degrees  of  freedom. 

192  P.  Ehrenfest,  Verhandl.  d.  deutsch.  physikal.  Ges.  15,  451  (1918). 

193  According  to  note  48,  the  quantum  canonical  distribution  function  is 


166  THE  QUANTUM  THEORY 

and  the  mean  energy  is 


~kT 


If  we  here  set  all  pn's  =  1,  and  if  for  En  we  substitute  the  value  E(n 
from  (80),  there  follows  for  the  mean  rotational  energy  of  a  molecule 


o 
and  for  the  heat  of  rotation  of  hydrogen  we  get  the  expression 


19*  The  turning  impulse  (moment  of  momentum)  of  a  system,  the 
mass-points  of  which  possess  the  mass  mi,  the  velocities  vi,  and  the  dis- 
tances n  from  a  fixed  point  (say  the  origin  of  co-ordinates),  is  a  vector  of 
the  value 


In  the  present  case,  the  system  consists  only  of  the  two  atoms  (mass  M)  , 
which  rotate  around  a  circle  of  radius  r  with  the  constant  velocity 
v  =  r  •  2irv. 
Hence  here 

|U|  =p  =  2Mra  .  2™  =  J.  2w, 

where  J  =  2Mr*  is  the  moment  of  inertia. 
198  The  impulse  (or  momentum)  pi  corresponding  to  a  generalised  co- 

(3L 
ordinate  qi  is,  according  to  note  48,  defined  by  the  relation  pi  =  ^  .  .' 

where  qi  =  -jr,  and  L  is  the  kinetic  energy  of  the  system.     Now  here 

the  angle  of  rotation  $  is  chosen  as  a  generalised  co-ordinate.  But  the 
kinetic  energy  of  a  body  rotating  about  a  fixed  axis  is  known  to  be 
=  J  '  (moment  of  inertia)  x  (angular  velocity)2,  hence 

J/d<t>\*_J- 
L=2\-Tt)    ~2*- 
Hence 

])Q  =  ov   =  J<p  =  3  '  2irv. 


NOTES  AND  REFERENCES  167 

196  F.  Reiche,  Ann.  d.  Phys.  58,  657  (1919). 

197  The  best  curve  was  obtained  by  assigning  the  "  weight  "  In  to  the 
nth  quantum  state  of  rotation.      The  rotationless  state  (n  =  0)  thus 
receives  the  weight  zero,  i.e.  it  does  not  exist.     This  amounts  to  the 
same  thing  as  the  introduction  of  a  zero-point  rotation. 

198  E.  Holm,  Ann.  d.  Phys.  42,  1311  (1913). 

199  J.  v.  Weyssenhoff,  Ann.  d.  Phys.  51,  285  (1916). 

200  M.  Planck,  Ber.  d.  deutsch.  physikal.  Ges.  17,  407  (1915). 

201  S.  Rotszayn,  Ann.  d.  Phys.  57,  81  (1918). 

202  The  curve  is  not  drawn  by  Planck,  but  is  discussed  in  the  author's 
paper  cited  in  note  196. 

203  See  likewise  the  author's  paper  quoted  in  note  196. 

204  P.  S.  Epstein,  Ber.  d.  deutsch.  physikal.  Ges.  18,  398  (1916).    Of. 
also  Phys.  Zeitschr.  20,  289  (1919). 

203  N.  Bohr,  Phil.  Mag.  1913,  p.  857. 

206  During  "  regular  precession  "  the  top  turns  uniformly  about  its 
axis  of  symmetry  (axis  of  its  figure),  while  at  the  same  time  this  axis 
describes  a  cone  of  circular  section  about  an  axis  fixed  in  space. 

207  A  compilation  of  the  moments  of  inercia  of  the  hydrogen  molecule 
used  by  the  various  investigators  is  as  follows  : — 

.7.10«. 

Einstein-Stern 1-47 

Ehrenfest 0'69 

(2-214^ 
Reiche \  2-293  \  different  curves. 

12-095J 

Holm 1-36 

Weyssenhoff 0-34 

Rotszayn 2-12 

Epstein  (Bohr's  model)       .        .        .     2-82 

MSN.  Bjerrum,  Nernst  Festschrift  1912,  p.  90.     Bjerrum  did  not, 
by  the  way,  start  from  formula  (79),  but  calculated  with  the  values 

yn=  o~Tr>  s^uce>  following  a  proposal  of  H.  A.  Lorentz,  he  set  the  rota- 
tional energy  E^  equal  to  nhvn,  in  contrast  to  Ehrenfest's  formulation 

(78),  which  rests  on  a  sounder  basis. 

209  S.  P.  Langley,  Annals  of  the  Astrophysical  Observatory  of  the 
Smithsonian  Institution,  Vol.  I,  p.  127,  Plate  XX  (1900). 

210  F.  Paschen,  Wiedem.  Ann.  51,  1 5  52,  209  ;  53,  335  (1894). 

211  H.  Rubens,  Berl.  Ber.  1913,  p.  513. 

212  H  Rubens  and  E.  Aschkinass,  Wiedem.  Ann.  64,  584  (1898). 

213  If.  Rubens  and  G.  Hettner,  Berl.  Ber.  1916,  p.  167.      See  also 
G.  Hettner,  Ann.  d.  Phys.  55,  476  (1918). 

214  W.  Burmeister,  Ber.  d.  deutsch.  physikal.  Ges.  15,  589  (1913). 

215  Eva  v.  Bahr,  Ber.  d.  deutsch.  physikal.  Ges.  15,  710,  731,  1150 

(1M63<3f .  Lord  RayUigh,  Phil.  Mag.  34,  410  (1892).    Let  an  HC1  mole- 
cule,  for  example,  be  considered,  which  consists  of  a  positively  charged 


168 


THE  QUANTUM  THEORY 


hydrogen  atom  H+  and  a  negatively  chlorine  atom  Cl~  (see  Fig.  13).  Let 
its  centre  of  gravity  be  S,  and  let  a  be  the  distance  of  the  H+  atom  from 
S.  Let  the  line  joining  the  two  atoms  be  the  axis  of  x',  and  let  this  axis 
turn  in  the  positive  direction  about  S  at  the  rate  of  vr  revolutions  per 
second  with  respect  to  the  fixed  x-7/-system.  If,  now,  the  two  atoms 
vibrate  relatively  to  one  another  with  the  frequency  rfl  and  the  amplitude 
A,  then  the  x'  co-ordinate  of  the  H+  atoms  may  be  represented  thus 

x'  =  a  +  A  sin  (2irv0t). 

If  we  project  this  vibration  upon  the  fixed  co-ordinate  system,  it  follows 
that 

(x  =  x'  cos  (2*vJ)  =  o  cos  (2irvrt)  +  A  sin  (2trv0t)  cos  (2*V) 
\y  =  y'  sin  (2irvrt)  =  a  sin  (torvj)  +  A  sin  (2*V)  sin  (2W) 


FIG.  13. 
for  which  we  may  also  write 

fa  =  a  cos  (2»yrO  +  4  sin  2»(r0  +  ,r)t  +  4  sin  2*(v,  -  ,r)t 
|T/  =  a  sin  (2w^)  -  ^cos  21r(^0  +  ,r)i  +  ^  cos  2»(ir0  -  Vf)t  . 

From  the  point  of  view  of  the  system  at  rest  we  have  thus  three 
oscillations  : 

(a)  the  left-circular  oscillation 

x  =  a  cos  (2^)|^  h  ^   {requenc 
T/  =  o«in(2»x^)J 
(6)  the  left-circular  oscillation 


with  the  frequency  vu  +  vr 


NOTES  AND  REFERENCES 

(c)  the  right-circular  oscillation 

I  with  the  frequency  vn  - 1 


169 


~  sin  2 


217  E.  S.  Imes,  Astroph.  Journ.  50,  251  (1919). 

218  A.  Eucken,  Ber.  d.  deutsch.  phys.  Ges.  15,  1159  (1913).     Eucken  has 
here,  on  account  of  the  asymmetrical  form  of  the  hydrogen  molecule, 
assumed  two  different  moments  of  inertia 

</!  =  0-96  .  10-40,  and  Jt  =2-21  .  lO"40 

and  hence  obtained  two  different  series  of  numbers  giving  the  revolutions 
Vr  per  second,  cf  .  the  table  given  there.  See  also  the  table  in  Rubens  and 
Hettner,  loc.  cit.,  note  213. 

219  M.  Planck,  Ann.  d.  Phys.  52,  491  ;  53,  241  (1917). 

220  0.  Sackur,  Ann.  d.  Phys.  36,  958  (1911)  ;  40,  67  (1913). 

•  221  H.  Tetrode,  Phys.  Zeitschr.  14,  212  (1913)  ;  Ann.  d.  Phys.  38,  434 
(1912).  , 

222  W.  H.  Keesom,  Phys.  Zeitschr.  15,  695  (1914). 

2234.  Sommerfeld,  Vortrage  liber  die  kinetische  Theorie  der  Materie 
und  der  Elektrizitat.  Wolfskehl-Kongress  in  GQttingen  1913.  (Teubner, 
Leipzig  and  Berlin  1914),  p.  125. 

224  P.  Scherrer,  Gottinger  Nachr.  8  July,  1916. 

225  M.  Planck,  Berl.  Ber.  1916,  p.  653. 

226  W.  Nernst,  Die  theoretischen  und  experimentellen  Grundlagen  des 
ueuen  Warmesatzes.     (W.  Knapp,  Halle  1918),  pp.  154  et  seq. 

227  O.  Sackur,  Ber.  d.  deutsch.  chem.  Ges.  47,  1318  (1914). 

228  0.  Stern,  Phys.  Zeitschr.  14,  629  (1913)  ;  Zeitschr.  f.  Elektrochemie 
25,  66  (1919). 

229  For  what  follows  cf.  the  paper  by  O.  Stern  quoted  in  the  last  note. 
Further,  W.  Nernst,  Die  theoretischen  und  experimentellen  Grundlagen 
des  neuen  Warmesatzes.     (W.  Knapp,  Halle  1918),  Oh.  XIII. 

230  As  regards  this  and  the  following  chapter,  the  reader  is  referred  for 
more  exact  details  to  the  article  of  P.  S.  Epstein  in  the  Planck  number 
of  "  Naturwissenschaften  "  (1918,  p.  230). 

231  As  the  simplest  Tiwmson  atom,  we  are  to  imagine  a  sphere  of  radius 
a,  filled  with  the  unit  charge  e  of  posi- 

tive electrification,  of  space-density  p,  in 

the  middle  of  which  an  electron  with 

the  charge  -  e  rests.     This  structure  is 

externally    neutral.      If    we   draw   the 

electron  out  from  the  centre  to  a  distance 

r  (see  Fig.    14)  the  external   (shaded) 

hollow  sphere  exerts  no   force  on  the 

electron,  according  to  the  well-known 

laws  of  electrostatics.     The  inner  solid 

sphere  of  radius  r,  on  the  other  hand, 

acts  on  the  electron  just  as  if  its  total  „ 

charge  were  concentrated  at  the  centre. 

The  force  which  draws  the  electron  back  into  its  position  of  rest  is 


170  THE  QUANTUM  THEORY 

therefore, 


that  is,  it  is  proportional  to  the  distance  of  the  electron  from  its  position 
of  equilibrium. 

232  Cf.  also  P.  Drude,  Lehrbuch  der  Optik.     2.  Aufl.,  Chs.  V  and  VII 
(Hirzel  1906).     There  is  an  English  edition  of  this  work. 

233  Cf .  W.  Voigt,  Magneto-  und  Elektro-optik  (Teubner  1908). 

234  M.  Planck,  Ber.  d.  Berl.  Akad.  d.  Wiss.  1902,  p.  470;  1903,  p.  480; 
1904,  p.  740 ;  1905,  p.  382. 

238  H.  A.  Lorentz,  The  Theory  of  Electrons,  Chs.  Ill,  IV  (Teubner  1909). 

236  The  electron  oscillates,  when  bound  quasi-elastically,  according  to 

the  equation  of  motion  m— ?  =  -  fx,  if  we  restrict  ourselves  to  linear  os- 
cillations. Here  m  is  the  mass  of  the  electron,  x  is  its  distance  from  the 
position  of  rest,  and  /  is  a  factqr  of  proportionality.  The  solution  of  this 
differential  equation  is  represented  by  the  pure  harmonic  motion 

x  =  A  cos  (nt  +  8) 
where  the  frequency  is 

n  =  V  TO' 

The  frequency  n  is  therefore,  as  we  see,  independent  of  the  amplitude  and 
therefore  of  the  energy  of  vibration. 

237  The  frequencies  v  of  those  spectral  lines  of  luminous  hydrogen, 
which  are  included  under  the  name  "  Balmer  series,"  may  be  represented 
with  great  accuracy  by  the  following  formula  given  by  Balmer. 

n  =  3,  4,  5,  6  ...  oo  . 
\&'      n'/ 

N  is  here  a  constant,  the  so-called  Rydberg  number.  If  we  set  for  the 
current  number  n  the  values  3,  4,  5  ...  we  get  in  succession  the  fre- 
quencies of  the  red  line  of  hydrogen  (Ha),  the  green  line  (H^),  and  the 
blue  line  (Hy)  and  so  forth. 

238  J.  Stark,  Ann.  d.  Phys.  43,  965  (1914) ;  J.  Stark  and  G.  Wendt,  ibid., 
43,  983  (1914) ;  J.  Stark  and  H.  Kirschbawn,  ibid.,  43,  991 ;  43,  1017 
(1914) ;  J.  Stark,  ibid.,  48,  193,  210  (1915) ;  J.  Stark,  O.  Hardtke  and  G. 
Liebert,  ibid.,  56,  569  (1918) ;  J.  Stark,  ibid.,  56,  577  (1918) ;  G.  Liebert, 
ibid.,  56,  589,  610  (1918) ;  J.  Stark  and  0.  Hardtke,  ibid.  58,  712  (1919) ; 
J.  Stark,  ibid.,  58,  723  (1919). 

239  Cf.  H.  A.  Lorentz,  The  Theory  of  Electrons  (Teubner,  Leipzig  and 
Berlin  1909),  Ch.  III. 

2*0  H.  Geiger  and  Marsden,  Phil.  Mag.  April,  1913. 

241  E.  Rutherford,  Phil.  Mag.  21,  669  (1911). 

242  According  to  C.  G.  Darwin  [Phil.  Mag.  27,  506  (1914)],  the  radius 
of  the  nucleus,  taken  as  a  sphere,  is  in  the  case  of  gold  at  the  most 
=  3  .  10 -12  cms.,  in  the  case  of  hydrogen  at  the  most  =  2  .  10-13  cms. 


NOTES  AND  REFERENCES  171 

243,4.  van  den  Broek,  Phya.  Zeitschr.  14,  32  (1913). 

244  Cf.  note  247. 

243  N.  Bohr,  Phil.  Mag.  26,  1,  476,  857  (1913). 

2464.  Einstein,  Phys.  Zeitschr.  18,  121  (1917). 

247  The  quite  elementary  calculation  is  as  follows  :  let  an  electron  of 
charge  e  and  mass  m  rotate  around  a  nucleus  of  charge  E  =  ez  in  a 
circular  orbit  :  then  z  is  the  atomic  number  (for  hydrogen,  in  particular, 
z  =  1).  If  a  is  the  radius  of  the  circle,  v  the  velocity,  and  a>  the  angular 
velocity  (frequency  of  rotation)  of  the  electron  in  the  circular  orbit,  then 
the  condition  for  equilibrium  between  the  attraction  of  the  nucleus  and 
the  centrifugal  force  is 

e_  =  maw2    or    masw2  =  eE  =  e*z. 

According  to  Bohr's  second  hypothesis  the  moment  of   momentum 
p(  =  mva=ma?<u)  is  a  multiple  of  —  ,  hence 

ma*a  =  wA        (n  =  1,  2,  3  .  .  .). 

From  these  two  equations  for  a  and  u>  we  get  for  the  discrete  radii  of  the 
permissible  quantum  orbits 


and  the  corresponding  frequencies  of  rotation 
^BvWfm 

The  energy  (kinetic  +  potential)  is 

(aT7t\  «2tf 

-T)  -*"*•-*- 

therefore  the  discrete  quantum  values  of  the  energy  are 


*  Wn?    ' 

If,  in  this  expression,  we  set 


we  recognise,  that  W  is  a  function  of  «,  and  hence  of  v  =  £-.    The  energy 

of  the  electron  in  the  Rutherford  model  therefore  depends,  as  stated  in 
the  text,  on  its  frequency  of  rotation  v. 

If  the  electron  passes  from  the  ntb  to  the  sth  quantum  path,  then,  ac- 
cording to  Bohr's  third  hypothesis,  a  homogeneous  spectral  line  is  emitted 
of  frequency 

Wn  -  Ws    2irVw.22/l      1 


172  THE  QUANTUM  THEORY 

where 

N  =  Wm 

2*8  Cf .  note  237. 

2*9  It  is  of  historical  interest  to  note  that,  before  Bohr,  A.  E.  Hems  in 
1910  (Sitzungsber.  d.  Wiener  Akad.  10  March,  1910)  succeeded  in  repre- 
senting Rydberg's  number  in  terms  of  the  universal  constants  e,  h,  m ; 
his  result  differed  from  that  of  Bohr  only  by  a  factor  8.  He  deduced  his 
result  as  follows.  Starting  from  /.  J.  Thomson's  atomic  model,  which 
was  generally  accepted  at  that  time,  he  calculated  the  maximum  oscilla- 
tion-frequency (no.  of  revolutions)  »/max  of  the  electron  in  the  simplest 
atom  (hydrogen  atom)  for  the  case  when  this  atom,  provided  with  one 
energy-quantum,  was  circling  just  on  the  surface  of  the  positive  sphere. 
He  obtained 

4»Vw 
"max  =  — p — 

This  maximum  frequency  was  next  identified  by  Haas  with  the  series 
limit  (n  =  oo  )  in  Balmer's  formula 


Then  it  follows  that 


which  is  a  value  8  times  greater  than  JVjjohr-  Haas  used  this  relation  to 
calculate  from  the  three  quantities,  the  Bydberg  number  N,  Planck's 

constant  h,  and  the  ratio  ~  ,  all  of  which  he  assumed  known,  the  charge 
m 

e  of  the  electron.  In  consequence  of  the  factor  8  he  obtained  the  value 
e  =  3'18  .  10~10,  a  value  that  is  too  small  according  to  our  present  know- 
ledge, but  which  agreed  well  with  the  measurements  of  J.  J.  Thomson  and 
H.  A.  Wilson,  which  were  available  at  that  time. 

230  Th.  Lyman,  Phil  Mag.  29,  284  (1915). 

251  F.  Paschen,  Ann.  d.  Phys.  27,  565  (1908). 

252.4.  Fowler,  Month.  Not.  Roy.  Astron.  Soc.  73,  Dec.  1912. 

233  F.  Paschen,  Ann.  d.  Phys.  27,  565  (1908). 

23*  E.  C.  Pickering,  Astroph.  Journ.  4,  369  <1896)  ;  5,  92  (1897). 

238  E.  J.  Evans,  Nature,  93,  241  (1914). 

236  W.  Kossel,  Ann.  d.  Phys.  49,  229  (1916)  ;  Die  Naturwissenschaften 
7,  339,  360  (1919). 

237  L.  Vegard,  Verhandl.  d.  deutsch.  physikal.  Ges.  19,  344  (1917). 

238  A.  Sommerfeld,  Atombau  und  Spektrallinien.     (An  English  edition 
translated  from  the  3rd  German  edition  (1922)  is  being  prepared   by 
Messrs.  Methuen  &  Co.,  Ltd.) 

259  R.  Ladenburg,  Die  Naturwissenschaften  8,  5  (1920). 
280  .4.  Sommerfeld,  Ann.  d.  Phys.  51,  1  (1916). 


NOTES  AND  REFERENCES  173 

281  Expressed  in  terms  of  polar  co-ordinates  the  kinetic  energy  L  as- 
sumes the  well-known  form  : 

L  =  «(^  +  rV). 

In  it,  m  denotes  the  mass  of  the  electron,  the  dots  represent  differentia- 
tion with  respect  to  the  time.  The  impulses  pr  and  p^  are  then  defined  as 
follows  (see  note  48)  : 

pr=?>L  =  mr  ;  p.  =  !%L  =  mr^. 

3r'  ^ 

262  Only  when  each  impulse  _pf  depends  solely  on  the  corresponding 
2j  (or  when  it  is  a  constant),  and  when,  in  addition,  the  limits  of  the 
phase-integral  are  independent  of  the  g/s,  does  the  phase-integral  work 
out  to  a  constant.     This  is  by  no  means  the  case  for  any  arbitrary  choice 
of  the  co-ordinate-system. 

263  P.  S.  Epstein,  Ann.  d.  Phys.  50,  489  ;  51,  168  (1916). 

264  K.  Schwarzschdd,   Sitzungsber.  d.  Berl.  Akad.  d.  Wiss.  4.  Map 
1916.       * 

263  A.  Einstein,  Verhandl.  d.  deutsch.  physikal.  Ges.  19,  82  (1917). 

266  M.  Planck,  Verhandl.  d.  deutsch.  physikal.  Ges.  17,  407,  438  (1915)  ; 
Ann.  d.  Phys.  50,  385  (1916). 

267  The  semi-major  axis  of  the  ellipse,  which  is  characterised  by  the 
values  n  and  n',  here  has  the  value 


The  ratio  of  the  axis  is 

b  _      n 
a~  n  +  n'' 
We  see  that  n'  =  0  corresponds  to  the  case  of  Bohr's  circular  orbits. 

268  The  energy  of  the  electron  moving  in  the  Kepler  ellipse  (n,  n')  here 
has  the  value 

=  _    2TrVsaw    _  _       Nhz* 

h\n  +  n'Y       ~  (n  +  n')a' 

The  series  formula  (102)  of  the  text  then  follows  from  Bohr's  Law  of 
Frequency 


269  If  account  is  taken  of  the  influence  of  relativity,  the  series  formula 
for  the  spectra  of  the  hydrogen  type  become  to  a  first  approximation 

V   =    VQ    +    Vl 

where 


174  THE  QUANTUM  THEORY 

In  these  expressions  the  symbols  N  and  a  have  the  following  meaning : 
N=      2fe*mo        a  =  ^f;  a2  is  of  the  order  5-3.10-5 


m0  is  the  mass  of  the  electron  at  vanishingly  small  velocities. 

Hence  whereas  the  first  term  i/0  gives  the  old  formula,  which  was 
obtained  by  neglecting  the  influence  of  relativity,  the  small  additional 
term  vl  represents  the  influence  of  relativity.  As  we  observe,  vl  does  not 
only  depend  on  the  quantum  sums  s  +  s'  and  n  +  n',  but  also  on  the 
individual  values  s,  s',  n,  n'.  This  member,  vv  is  thus  responsible  for  the 
fine-structure. 

270  If  we  apply  the  formula  of  the  preceding  note  to  Ha,  we  have  to  set 
z  =  1,  s  +  s'  =  2,  n  +  n'  =  3.  We  then  get 


l       s'      1      n'~\ 

+r  i  +  * 

24  84     J 


n 

AVjj 

j 

1  [- 

-i  1 

r  1  1 

t?  *>  * 
i  i 

n 

r                             c 

I 

1  i 

1  "i 

* 


Pia.  15. 


Corresponding  to  the  possibilities  of  partition 


and 


2  +  01  circle 
1  +  I/  ellipse 


2  ^^  orbits 


n  +  ri  =  3  =  3  +  0^  circle    \ 

=  2  +  1  L  ellipse  L  3  initial  orbits 
=  1  +  2]  ellipse  ) 

(for  dynamical  reasons  the  azimuthal  quantum  number  n  cannot  under 
normal  conditions  assume  a  zero  value),  we  should  expect  2.3  =  6 
possibilities  of  production  and  hence  6  components  of  the  fine-structure 
of  Ha..  One  of  these  components,  however,  namely,  the  one  correspond- 
ing to  the  transition  of  the  electron  from  the  circle  (n  =  3,  n'  =  0)  to  the 
ellipse  (s  =  1,  s'  =  1)  does  not  present  itself  under  normal  conditions,  as 
follows  from  the  "  Principle  of  Selection  "  enunciated  by  Rubinowicz 


NOTES  AND  REFERENCES  175 

and  Sommerfeld  (see  Chapter  VI,  §9).  Hence  5  components  of  the 
fine-structure  remain  ;  their  position  is  exhibited  in  Fig.  15. 

As  we  see,  the  5  components  arrange  themselves  into  two  main  groups, 
containing  3  and  2  members,  respectively.  The  «'  missing  "  line  Ila  is 
dotted  in.  The  distance  AVH  between  Ia  and  IIa,  Ib  and  lib,  Ic  and  IIC  is 
called  the  "  theoretical  hydrogen  doublet." 

According  to  the  above  formula  the  frequency-number  of  the  line 
la  (3,  0-»2,  0)  is 


The  frequency-number  of  the  line  Ha  (3,  0—  >1,  1)  is 


Thus 

A"H  =  */J,,  -  "/a  =  ^=1-095.1010 

corresponding  to  AA.H  =  0'157A. 

The  hydrogen-doublet  actually  observed  is  measured  from  about  the 
middle  of  Ia  and  It,  to  the  middle  of  lib  and  IIC,  owing  to  the  absence  of 
IIa.  This  leads  to  the  value  0'8AA.H,  that  is,  to  0-126A. 

According  to  a  principle  of  correspondence  enunciated  by  Bohr  (see 
Chapter  VI,  §  9),  as  a  result  of  which  the  azimuthal  quantum  number  can 
only  vary  by  +  1,  the  components  Ib  and  IIC  are  also  absent. 

271  F.  Paschen,  Ann.  d.  Phys.  50,  901  (1916). 

272  From  formula  (97)  of  the  text  we  get  for  the  two  Rydberg  constants 
for  hydrogen  and  helium  : 


Moreover,  according  to  note  269,  we  get  the  third  formula  giving  the 
value  of  the  constant  for  the  fine-structure  : 


From  the  first  two  relations,  by  using  MHO  =  &Ma,  we  get 

ma  ,   __M^ 

~ 


176  THE  QUANTUM  THEORY 

and  hence 

_L  =  JL     NH-*NH* 

m0c      MHC  '   Nne  -  Na  ' 

The  two  Rydberg  numbers  NH  and  NHC  have  been  measured  by  Pastfien 
with  great  accuracy  : 

Nil  =  (109677-691  +  0'06)  .  c 
Nue  =  (109722-144  ±  0'04)  .  c. 

Moreover,   —£  —  =  F  is  the  electrochemical  equivalent  (Faraday'*  num- 

MH  .  c 
ber),  that  is,  the  charge  which,  in  electrolysis,  accompanies  one  gramme- 

atom  (i.e.  N  =  —  -  atoms).    This  number  has  the  value 
F  =  9649-4  electromagnetic  units. 

If  we  insert  the  three  values  of  Na,  NH*  and      *  -  in  the  relation  above 

MH.C 
deduced,  we  get 

JL  =  1-7686  .  107  electromagnetic  units, 
m0c 

a  value  which  agrees  very  well  with  those  values  of  this  quantity  which 
were  obtained  by  direct  methods  (deflection  of  the  cathode-  and  /5-rays  in 
the  electric  and  magnetic  field).  Let  us  now  write 


or,  using  the  value  of  -       given  above, 
MH 

^m^  =  3 
h»          4 

The  right-hand  side  of  this  equation  is  known.     If  we  combine  with  it 

— 
m0c 

O       /»2 

«  =  —  =  7-290.  10-3 

which  follows  from  Paschen's  measurements  of  the  fine-structure  in  the 
case  of  helium,  we  have  three  equations  in  three  unknowns  e,  mg,  h. 
From  them  we  get 

e  =  (4-766  ±0-088).  10  -10 

h  =  (6-526  +  0-200)  .  10  -27. 

According  to  Sommerfeld  it  is  more  advantageous  to  use  Millikan's  value 
for  e.  We  then  get 

fe  =(4-774  ±0-004).  10  -10 
{  h  =  (6-545  ±  0-009)  .  1Q-87 
U  =  (7-295  ±  0-005)  .  10-». 


NOTES  AND  REFERENCES  177 

273  K.  Glitscher,  Ann.  d.  Phys.  52,  608  (1917). 

274  A.  Lande,  Phys.  Zeitschr.  20,  228  (1919)  ;  21,  114  (1920). 

278  Cf.  A.  Sommerfeld,  Atombau  und  Spektrallinien.    Ch.  IV,  §  6. 

276  P.  S.  Epstein,  Ann.  d.  Phys.  50,  489  (1916). 

277  P.  Debye,  Gottinger  Nachr.  3  June,  1916. 

278/1.  Sommerfeld,  Phys.  Zeitschr.  17,  491  (1916).     Cf.  also  Atombau 
und  Spektrallinien.     Ch.  VI,  §  5. 

279  F.  Paschen  and  E.  Back,  Ann.  d.  Phys.  39,  897  (1912)  ;  40,  960 
(1913). 

2804.  Rubinowicz,  Phys.  Zeitschr.  19,  441,  465  (1918). 

281  N.  Bohr,  On  the  Quantum  Theory  of  Line-spectra.    Parts  I  and  II. 
D.  Kgl.  Danske  Vidensk.  Seisk.  Skrifter,  Naturvidensk.  og  Mathem.  Afd. 
8,  Baekke  IV,  1.     Kopenhagen  1918. 

282  The  number  of  revolutions  of  the  electron  per  second  in  the  sth 
quantum  circle  of  Bohr  is,  in  the  case  of  hydrogen,  according  to  note  247  : 


On  the  other  hand,  it  follows  from  formula  (93)  of  the  text,  if  we  take  s 
considerably  greater  than  1  (high  quantum  numbers),  and  n  =  s  +  1 
(transition  between  neighbouring  circles),  that 


283  P.  S.  Epstein,  Ann.  d.  Phys.  58,  553  (1919). 

284  H.  A.  Kramers,  Intensities  of  Spectral  Lines.      D.  Kgl.  Danske 
Vidensk.  Selsk.  Skrifter,  Naturvidensk.  og  Mathem.     Afd.  8,  Raekke  III, 
3.     Kopenhagen  1919. 

283.4.  Sommerfeld  and  W.  Kossel,  Ber.  d.  deutsch.  physikal.  Ges.  21, 
240  (1919). 

286  J.  Franck  and  G.  Hertz,  Phys.  Zeitschr.  20,  132  (1919)  ;  in  which 
references  are  also  given.     Cf  .  also  J.  Franck  and  P.  Knipping,  Phys. 
Zeitschr.  20,  481  (1919)  ;  J.  Franck,  P.  Knipping  and  Thea  KrOger,  Ber. 
d.  deutsch.  physikal.  Ges.  21,  728  (1919). 

287  J.  Tate  and  Foote,  Phil.  Mag.  July,  1918. 

288  References  are  given  in  the  report  by  J.  Franck  and  G.  Hertz,  men- 
tioned in  note  286. 

289  A.  Einstein,  Phys.  Zeitschr.  18,  121  (1917).     Let  us  consider  the 
two  quantum  states  (1)  and  (2)  of  the  atom,  with  the  energies  El  and  Ez 
(.E2  >  -EJ).     The  number  of  transitions  2  -»  1  which  take  place  in  the 
time  dt  owing  to  radiation  is  then,  according  to  Einstein,  NiA2ldt,  in 
which  N9  is  the  number  of  atoms  in  the  state  2,  and,  therefore,  accord- 
ing to  note  48 

~  &. 


N,,  =  Nw, 

N  being  the  total  number  of  atoms.    4al  is  a  factor  of  proportionality. 
12 


178  THE  QUANTUM  THEORY 

The  introduction  of  external  monochromatic  radiation  of  frequency  «/ 
and  intensity  K,,  firstly  brings  about  positive  absorption,  that  is,  transi- 
tions l->2.  The  number  of  these  in  the  time  dt  is,  according  to 
Einstein,  JVj.B12K,,,  in  which  .B12  is  a  factor  of  proportionality,  Nl  is  the 
number  of  atoms  in  the  state  1,  and  hence 

5 

jY,  =  NCpje    M. 

Secondly,  the  external  radiation  also  effects  transitions  2  -»  1  (nega- 
tive absorption).  The  number  of  these  that  occur  in  the  time  dt 
=  -ZV-j-B^Kj/,  where  jB2]  is  a  factor  of  proportionality.  When  the  energy 
exchange  is  in  equilibrium  the  number  of  transitions  2  —  >•  1  must  be 
equal  to  the  number  of  transitions  1  —  >  2,  hence 


~  M' 


i.e. 


When  the  temperature  increases  indefinitely,  K*  must  also  increase  to 
infinitely  great  values  ;  from  this  it  follows  that 

=  1. 

Finally,  if  we  set  -^p-  =  A  for  shortness,  we  get  the  relation  given  in  the 
text: 

•^  A 


290  Of.  the  resume*  by  E.  Wagner,  Phys.  Zeitgchr.  18,  405,  432,  461, 
488  (1917). 

291  O.  Moseley,  Phil.  Mag.  26,  1024  (1913) ;  27,  703  (1914). 

292  W.  Kossel,  Verhandl.  d.  deutsch.  physikal.  Ges.  16,  898,  953  (1914)  ; 
18,  339  (1916). 

293  A.  Sommerfeld,  Ann.  d.  Phys.  51,  125  (1916) ;  Phys.  Zeitschr.  19, 
297  (1918).      Of.  also  Atombau  und  Spektrallinien.      Oh.  Ill,  Ch.  IV 
§  4,  Ch.  V  §  5. 

29*  L.  Vegard,  Verhandl.  d.  deutsch.  physikal.  Ges.  1917,  pp.  328,  344  ; 
Phys.  Zeitschr.  20,  97,  121  (1919). 
298  P.  Debye,  Phys.  Zeitschr.  18,  276  (1917). 

296  J.  Kro6,  Phys.  Zeitschr.  19,  307  (1918). 

297  A.  Smekal,   Wiener  Ber.  Ila  127,  1229  (1918) ;  128,  639  (1919) ; 
Verhandl.  d.  deutsch.  physikal.  Ges.  21,  149  (1919).     Of.  also  A.  Smekal 
and  F.  Reiche,  Ann.  d.  Phys.  57,  124  (1918). 


NOTES  AND  REFERENCES  179 

298  W.  Kossel,  Zeitschr.  f.  Physik  1,  119  (1920). 

299  Since  the  L-ring  consists  of  several  electrons,  we  take  the  expression 
"  elliptic  motion  "  to  mean  the  following  type  of  motion  :  each  electron 
independently  describes  an  elliptic  path  about  the  nucleus,  whereby  the 
electrons  are  at  each  moment  situated  at  the  corners  of  a  regular  polygon 
which  shares  in  the  motion  of  the  electrons,  alternately  contracting  and 
expanding  during  this  motion  ("  elliptical  associates"),  cf.  Sommerfeld. 
Atombau  and  Spektrallinien. 

3004.  Stnekal,  Wiener  Ber.  Ha  128,  639  (1919). 

301  M.  Born  and  A.  Lande,  Berl.  Akad.  Ber.  1918,  p.  1048  ;  Verhandl, 
d.  deutsch.  physikal.  Ges.  20,  202,  210  (1918) ;   M.  Born,  ibid.,  20,  230 
(1918) ;  Ann.  d.  Phys.  61,  87  (1920). 

302  A.  Lande,  Verhandl.  d.  deutsch.  physikal.  Ges.  21, 2, 644, 653  (1919) ; 
Zeitschr.  f.  Phys.  2,  83  (1920).    Cf.  also  A.  Lande  and  E.  Madelung, 
Zeitschr.  f.  Phys.  2,  230  (1920). 

303  W.  Kossel,  Ann.  d.  Phys.  49,  229  (1916). 
3M  P.  Debye,  Munch.  Akad.  Ber.  9  Jan.  1915. 

305  P..  Scherrer,  Die  Botationsdispersion  des  Wasserstoffs.    Dissertation, 
Gottingen,  1916. 

306  G.  Laski,  Phys.  Zeitschr.  20,  269,  550  (1919). 

307  Cf.,  for  example,  A.  Sommerfeld,  Atombau  und  Spektrallinien,  Ch. 
IV,  §  6. 

308  Langmuir,  Journ.  Amer.  Chem.  Soc.  34,  860  (1912) ;  Zeitsohr.  f. 
Electrochemie  23,  217  (1917). 

309  Isnardi,  Zeitschr.  f.  Elektrochemie  21,  405  (1915). 

310  /.  Franck,  P.  Knipping  and  Thea  KriLger,  Ber.  d.  deutsch.  physikal. 
Ges.  21,  728  (1919). 

310a  Planck  has  made  an  attempt  to  alter  Bohr's  model  in  such  a  way 
that  the  right  heat  of  dissociation  results.  See  M.  Planck,  Berl.  Akad. 
Ber.  1919,  p.  914.  Cf.  also  H.  Kallmann,  Dissertation,  Berlin  1920. 

311  W.  Lenz,  Ber.  d.  deutsch.  physikal.  Ges.  21,  632  (1919). 
3124.  Sommerfeld,  Ann.  d.  Phys.  53,  497  (1917). 

313  F.  Pauer,  Ann.  d.  Phys.  56,  261  (1918). 

314  G.  Laski,  see  note  314. 

315  M.  Pier,  Zeitschr.  f.  Elektrochemie  16,  897  (1910). 

316  K.  Schwarzschild,  Berl.  Akad.  Ber.  1916,  p.  548. 

317  H.  Deslandres,  Compt.  Rend.  138,  317  (1904). 

318  T.  Heurlinger,  Phys.  Zeitschr.  20,  188  (1919) ;  Zeitschr.  f.  Physik 
1,  82  (1920). 

319  W.  Lenz,  see  note  311. 

320  A  different  view  is  upheld  by  /.  Burgers  (Versl.  K.  Ak.  van  Wet. 
Amsterdam  26,  115,  1917),  in  which,  also,  jumping  electrons  produce  the 
middle  line  in  the  infra-red  of  band.     In  contrast  with  Schwartschild  and 
Lenz,  Burger  assumes  that  the  motion  of  the  electrons  is  influenced  by 
the  rotation  of  the  molecule.     The  energy  of  the  system  is  then  not  com- 
posed additively  of  the  energy  of  the  electrons  and  the  rotational  energy 
of  the  molecule,  but  a  third  term  has  to  be  added,  which  is  due  to  the 
Coriolis  force  of  the  rotating  system. 


180  THE  QUANTUM  THEORY 

321  T.  Heurlinger,  see  note  318. 

322  F.  Reiche,  Zeitschr.  f.  Physik  1,  Heft  4,  283  (1920). 

323  E.  S.  Imes,  Astrophys.  Journ.  50,  251  (1919). 

324  A.  Kratzer,  Dissertation,  Munchen  1920. 

323  In  the  case  of  the  gases  investigated  by  Imes,  namely  HC1,  HBr,  and 
HP,  the  following  moments  of  inertia  were  found : 

/HOI  =  2'6*  •  10- 40  ;        JHBr  =  3-27  .  10 -40 ;        JHF  =  1'37  .  10-40. 


INDEX 


Absorption  band,  edge  of,  21,  112. 

—  continuous,  78. 
Acoustic  vibrations,  157. 
Action,  quantum  of,  9,  27. 
Atomic  heat,  29. 

—  numbers,  86,  110. 
Avogadro's  constant,  12,  29,  163. 

—  Law,  12. 

Azimuthal  quantum  number,  92. 

B 

Back,  22. 

Balmer,  90. 

Barkla,  20,  109. 

Beck,  25. 

Benedicks,  146. 

Bergen,  106. 

Bergmann  series,  89. 

Bishop,  106. 

Bjerrum,  75,  123. 

Bohr,  75. 

Bohr's  model  of  the  atom,  86,  119. 

—  principle  of  analogy  or   corre- 

spondence, 100. 

selection,  98. 

Boltzmann,  4,  13. 

Born,  38,  42. 

Bragg,  W.  H.  and  W.  L.,  109. 

Bravais,  37. 

Bremsstrahlung,  22,  109. 

Broek,  van  den,  86. 

Broglie,  de,  21. 

Bunsen,  109. 

Burgers,  179. 


Canal  rays,  142. 
Cavity  radiation,  law  of,  3. 
Characteristic    Bontgen     radiation, 
109. 


Chemical  constant,  81. 
Compressibility,  33. 
Conductivity,  thermal,  60. 
Coulomb,  12. 


Davis,  106. 

Debye,  38,  39,  58. 

Debye's  formula,  40. 

Degeneration  of  gases,  79. 

Deslandres,  122. 

Dessauer,  22. 

Diamagnetism,  25. 

Diamond,  33. 

Dispersion,  47. 

Displacement  Law,  Wien's,  4. 

Distribution  numbers,  114. 

—  of  velocities,  Maxwell's  Law  of, 

5. 

Dix&n,  100. 
Doppler,  4. 
Drude,  61,  84,  17. 
Duane,  22. 
Dulong  and  Petit's  Law,  29,  30. 


Ehrenfest,  71. 

Einstein,  1,  15,  16,  30,  107. 

—  functions,  31,  32,  51. 
Einstein's     hypothesis     of     light- 
quanta,  16,  17,  107. 

—  Law  of  the  photo-electric  effect, 

20. 
Elastic  collisions,  104. 

—  spectrum,  42. 

Electron  theory  of  metals,  61. 
Emissivity,  2. 
Entropy,  8,  151. 
Epstein,  75,  98. 
Equipartition,  13. 
Eucken,  56,  61,  69. 


181 


182 


THE  QUANTUM  THEORY 


P 


Fine-structure,  Sommerf eld's  theory 

of,  94. 

—  of  the  Rontgen  rays,  113. 
Fokker,  15. 
Foote,  105. 
Franck,  22,  102. 
Friman,  109. 


Oeiger,  85. 
Gibbs,  13,  26. 
Glitscher,  97. 
Ooiicher,  106. 
Gramme-molecule,  145. 
Grilneisen,  58. 


Haas  de,  25. 

Haber,  46,  66,  155. 

Hamilton,  15. 

Hamilton's  principle,  27. 

Hauer,  von,  67. 

Heat  theorem,  Nernsfs,  35. 

Helium,  97. 

Helmholtz,  160. 

.ZTerte,  22,  102. 

Herzfeld,  66. 

Heurlinger,  122. 

Iftmse,  69. 

floo&e's  Law,  59. 

flop/,  15,  56. 

Hughes,  106. 

fltiK,  22. 

Hunt,  22. 

Hydrogen  type  of  series,  88. 


Imes,  123. 

Impulse  or  momentum,  26. 

—  radiation,  22,  109. 

Infra-red  dispersion  frequencies,  4i 

Intensity,  2. 

Isnardi,  120. 


Jeans,  15. 


K 

Kammerlingh-Onnes,  64. 
Karmdn,  38. 
Kepler  ellipses,  95. 
Kinetic  radiation,  23. 
Kirchhoff,  3. 
.KbsseZ,  91,  102,  105. 
Kramers,  102. 
Kratzer,  124. 
K-rings,  111. 
£roo,  111,  115. 
K-series,  21. 
Kurlbaum,  9. 


Ladenburg,  91. 

Langmuir,  120. 

Losfci,  119. 

Lattice  theory  of  atomic  heats,  42. 

Lowe,  35 

Law,  Avogadro's,  12. 

—  equipartition,  13. 

—  of  cavity  radiation,  3. 

—  of  radiation,  Planck's,  10. 
---  Bayleigh's,  10,  14. 
---  Wieris,  5,  7. 

—  Stefan  and  Boltzmann,  5. 
L-doublets,  114. 

Lechner,  56. 
Lees,  64. 
Lenard,  19. 
Lews,  120,  122. 
Light-quanta,  Einstein's,  16. 
Lindemann,  33,  64,  66. 
Lorentz,  15,  24,  61,  161. 
Lorenz,  62. 
L-rings,  111. 
Lummer,  2,  3,  4. 

M 


,  33. 
Marsden,  85. 
Maxwell,  5,  13. 
Mean  atomic  volume,  53. 
Meissner,  64. 

Metals,  electron  theory  of,  61. 
Michelson-Morley,  1. 
Millikan,  12,  20. 
Molecular  heat,  53. 
Momentum,  26,  166. 
—  moment  of,  72,  166. 
Moseley,  110. 


INDEX 


188 


N 

Nernst,  9. 

—  and  Lindemann's  formula,  37. 
Nernst's  heat  theorem,  35. 
Nichols,  33. 


Orbits,  allowable,  87. 
Ortvay,  152. 


Paramagnetism,  25. 

Parhelium,  97. 

Paschen,  6,  9,  89. 

Pauer,  120. 

Phase-integral,  92. 

Phase-space,  26,  92,  94,  136. 

Photo -electrons,  20,  141. 

Pier,  121. 

Planck,  6,  15,  24,  27,  91,  117. 

Poisson's  ratio,  41. 

Polarisation,  168. 

Precession,  167. 

Pringsheim,  2,  4. 

Probability,  thermodynamic,  8. 

Q 

Quanta,  energy-,  7. 
Quantum  of  action,  9. 


Radial  quantum  number,  92. 
Ratnowski,  58. 
Bayleigh,  10. 
Reflection,  metallic,  45. 
Relativity,  95,  96. 
Residual  rays,  35,  149. 
Resonance  lines,  104. 
—  potentials,  105. 
Resonators,  Planck's,  6. 
Bice,  22. 
Richards,  58. 
Richarz,  25. 
Riecke,  61. 

Rontgen  radiation,  21. 
Rotation  spectra,  76. 
Rotszayn,  75. 
Rubens,  9,  33. 

Rubinowicz'  principle  of  selection, 
98,  99,  174. 


Rutherford,  84,  171. 
Rydberg,  90,  110. 


Sackur,  80. 
Sadler,  21. 
Scheel,  69. 
Scherrer,  79,  119. 
Schrodinger,  57. 
Schwarzschild,  93,  121. 
Selection,  principle  of,  98. 
Sommerfeld,  27,  91,  94,  102,  120. 
Stark,  23,  66. 
—  effect,  98. 
Stefan,  4. 

Stefan-Boltzmann,  Law,  5. 
Stern,  25,  70,  81. 
Stokes,  19. 
Sutherland,  33. 


Thirring,  55. 

Thomson,  J.  J.,  19,  63,  84. 

—  atom,  169. 


Vegard,  91. 

117. 


w 

Wagner,  21,  22. 
Warburg,  9,  23. 
Weber,  30. 
Webster,  22. 
Weyssenhof,  von,  74. 

Wien,  3,  4. 

Wien's  displacement  Law,  4. 
—  Law  of  radiation,  5,  7. 
WW/,  9. 


Zeeman,  84. 

—  effect,  98,  120. 


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